What do we learn about male and female roles in Greek society from the Odyssey? How did you come to those conclusions?

We learn a few things about the roles of men and women in ancient Greek society from the Odyssey. One thing that is very apparent is that men were expected to be the heads of households and that women were expected to have a subordinate role. With Odysseus absent, his son Telemachus has assumed the role of master of the house. He has the power to dictate even what his mother does. For instance, in Book I Penelope asks the Phemios to stop singing the about the exploits of the heroes of the Trojan War. She asks him to sing of something else as the song reminds her of her absent husband. Telemachus makes it clear in his response just what he thinks of his mother's request. He tells her, "go to your room and occupy yourself with your own affairs, weaving and spinning and tell your maids to get busy. Poetry is the concern of men, particularly of me since I hold authority in the house." Telemachus tells his mother to attend to her weaving on several occasions, notably during the archery contest that Penelope herself suggested. He states that weapons are for men to be concerned about and not for women. This demonstrates a clear division of social spheres between men and women.
The issue of marital fidelity as related to men and women is also brought up in the epic poem. There is a clear double standard on this issue throughout the Odyssey. Penelope, a true model of the virtuous wife, remains faithful to her husband during his long absence. She fends off the persistent advances of her many suitors who in turn are killed for their lack of respect for Odysseus' property. Meanwhile, Odysseus carries on several affairs, such as with Calypso and Kirke, without judgment or consequence. It is clear that women were expected to remain loyal wives despite the circumstances, while men could have their own affairs.
https://www.gvsd.org/cms/lib02/PA01001045/Centricity/Domain/559/Gender%20roles%20in%20the%20Odyssey.pdf