DFA which accepts all strings starting and ending with 'aa'
One solution could be putting the final state right after the first two states.
From there, if an $a$ is read you stay in the final state, otherwise you move to a state analogous to $q_2$; from there, with two $a$s you return to the final state and with a $b$ you go back to $q_2$.

In short, just picture ($q_1, a$) going to $q_4$ instead of $q_2$. The rest should go unchanged.

This way, the automata should accept strings such as 'aa', 'aaa' and so on. And of course if it reads a 'b' it'll need two more 'a's to return to the final state.