Nondeterministic automaton accepting only words whose first letter is identical to their next to last letter
Hint: you want to make states $x/y/z$  which mean that $x$ is the first symbol of the word, $y$ is the second to last symbol seen, and $z$ is the last symbol seen.

I will show you half of the final machine, the one for words starting with $a$. Note that "!a" here means any symbol not equal to a, or no symbol at all.