Aumerle is the Duke of York's son and a bit of a puzzle in the play. He's one of the only characters who refuses to fall in neatly with Henry Bolingbroke's plans, even though he's also the only man to accompany Henry "to the next highway," where he'll take off for his banishment. This seems like a gesture of friendship, but it becomes clear later, when he plots against Henry (now king) that his feelings for his cousin are more complex than we'd thought. One reason is loyalty: he's Richard's man. Still, once York discovers his plot against Henry, he begs for forgiveness and ends up being King Henry's man too.
Aumerle's relationship with his father provides a foil to Gaunt's relationship with Henry. Just as Henry inherited and intensified Gaunt's bravery and bluntness, Aumerle inherited his father's philosophical obsession with loyalty. But where York pays lip service to loyalty and flips when Henry assumes power, Aumerle refuses to accept the new king. Instead, he participates in a plot against Henry.
York's rejection of Aumerle on the grounds of his disloyalty is surprising, coming from someone as wishy-washy as him. When York begs King Henry to kill his son, it's clear that something is deeply wrong, because Henry's new reign as king has pitted fathers and sons against each other, which hints at the civil wars that are going to break out in Henry IV Part 1. On the other hand, we can't necessarily blame Henry for all the domestic drama, because family members have been going at it (and even killing each other off) since long before this play started.