So, I've decided to create the mother of all walls of text in order to explain everything that TWSITD did in the long history of Fodlan.
Starting from the moment Sothis reached Fodlan. Now, it's not exactly hidden that TWSITD despise her. Considering that Rhea is the one that tells us about this, we have to take her statements with a grain of salt, as while she's definitely honest, that doesn't mean that her opinion isn't biased. It's possible that the Agarthans lived in Fodlan before Sothis came, as they refer to her as "Fell Star" (implying they saw her fall), and, although this is outside of the fourth wall, Sothis in real mythology is the greek name of the Egyptian goddess who was the incarnation of Sirius, a star that was considered a bad omen for the greeks, and Thales, Solon and Kronya are all greek names or still related to greek. Putting this in the FE perspective, the Agarthans saw Sothis as a foreign goddess that would bring bad luck.
Always according to Thales, after the Nabateans gave power to humans and they caused war, the Agarthans were actually forced by dragons to go underground, at difference of Rhea who simply said that they "retreated" (we don't know which one is correct, but the former would help to explain TWSITD's hatred for all dragons in general instead of just Sothis).
So, the Agarthans prepared their revenge. They manipulated Nemesis and convinced him that Fodlan needed to be "liberated" from the control of the Nabateans (hence why he later becomes the "King of Liberation"), and offered him their help in killing a sleeping Sothis. After killing her, TWSITD created the Sword of the Creator, crest stone and the crest of flames. After that, he and his elite warriors exterminated the children of the goddess who lived in Zanado, leaving only Rhea alive (and the other ones that weren't in Zanado at the time). After Nemesis got defeated, TWSITD got back his body, in order to try to resurrect him, as well as using his blood to give the Crest of Flames to who will later become their second Nemesis. The reason they manipulated Nemesis instead of doing that on their own was probably both to avoid any possible revenge and because they probably don't want dragon blood in their veins. I mean, they created crests and yet no Agarthan has them.
We don't know anything about what they did during most of Fodlan's history, but seeing Ionus' white hair, and Edelgard saying that the blood experiments were the dark secret of house Hresvelg in general, it's likely that they kept making experiments on this bloodline, hoping to create a second Nemesis.
1165: A plague in Fhirdiad threatens the population, but the real Cornelia prevents it from growing too big
1165-1176: Cornelia is killed and replaced by someone from TWSITD
1172: Hrym territory rebels, and house Ordelia helped them. After the rebellion was quelled by the empire, House Ordelia was effectively controlled by the empire, including TWSITD. Experiments on Lysithea, too.
1174: Arundel is killed and replaced by Thales (for reference, this is right after the insurrection of the seven and Edelgard's return to Enbarr after meeting Dimitri). Thales gets control of Hrym and house Ordelia. This or the next year are when the experiments on Edelgard's family start. They used Nemesis' blood to give the Crest of Flames to Edelgard, and tried to have her become a second Nemesis and manipulated her (this is probably where her intense hatred for the Children of the Goddess come from)
1176: Patricia, Edelgard's mother, starts plotting with Cornelia (who was replaced by TWSITD) a way to return to the empire. For still unknown reasons (we know that it was useful for TWSITD, but we don't know why), they helped her, as well as other kingdom nobles who didn't like Lambert, started plotting together the tragedy of Duscur.
1179: Monica gets abducted, and later killed. Tomas, after 40 years of work in Garreg Mach, comes back to house Ordelia, still controlled by TWSITD. He gets killed and replaced by Solon.
1180: Start of the game. From here on, I'll talk about the singular chapters.
Chapter 4: TWSITD discover that Byleth can wield the Sword of the Creator even without the crest stone, and realize that having a dangerous person on their enemy side means they're f*cked. If this is BE, they're doubly f*cked since they know very well that Edelgard hates them, and if she gains Byleth as an ally they're all dead. As such, most of their actions in part 1 are either to kill Byleth or prevent him from joining Edelgard. Btw, the attack on the tomb was very likely made so that Edelgard could get her hands on the Sword of the Creator, and fully become a second Nemesis.
Chapter 6: Flayn is kidnapped for her dragon blood, for TWSITD's blood experiments. They also use this occasion to make Kronya replace Monica and have her closely monitor Edelgard alongside Tomas (probably the entire reason they kidnapped and killed her the year before).
Chapter 8: Solon makes experiments on the population of Remire village. In BE, this has the very obvious purpose of connecting the Flame Emperor to such atrocious acts in order to prevent Byleth from ever joining the party that would cause such havoc in a village he cared about. In other routes, it's possible that they had the same goal in mind (as the Flame Emperor appears in every route trying to convince Byleth that he was not involved), but there's also the possibility that this experiment was made to transform people in demonic beasts (since the following chapter would be the first time we see people turning in Demonic Beasts), which is useful both for Edelgard's domination and extermination of dragons, and in order to bait Byleth and Jeralt in the ruined chapel. Flayn's blood may have been used here, as Claude tries to ask Flayn if she knew anything about the incident but she ran away before Claude could ask her.
Chapter 9: In the old chapel, Kronya transforms students into demonic beasts using the results of Solon's experiment in order to bait Byleth and Jeralt, and kill the latter. Thales also appears to save Kronya.
Chapter 10: Byleth, blinded by revenge against Kronya for killing his father, rushes in the sealed forest, which allows Solon to send him into a world of Eternal Darkness. This fails badly.
And I've finished. Everything they do post-timeskip is pretty clear and explained, so I don't think there's much to say here also because I'm lazy
@DDM/Dragon feel free to edit any grammar or logical mistake I made here. Actually, please do it.