Suppose we have a string homomorphism $\varphi: \Sigma \rightarrow \Sigma^*$. Consider the languages in $\varphi(\Sigma^*)$ whose letters are elements of $\varphi(\Sigma)$, so here I do not want to expand each $\varphi(\Sigma)$, but leave them in this abstract form, e.g., $\varphi(a)\varphi(b)\varphi(a)$ would be a word. So in fact for any language $L$, the language $\varphi(L)$ is the same as $L$, over a different alphabet. If $M$ was an automaton over $\Sigma$, then there is a natural automaton, $\varphi(M)$, over $\varphi(\Sigma)$ with the same number of states: from any given state after reading $\varphi(a)$, $\varphi(M)$ goes to whatever state $M$ would go after reading all letters of $\varphi(a)$.
For example, suppose that $M$ was counting the parity of $b$'s, that is, from state 0 it goes to state 1 iff the read letter is $b$, and vica versa. Let $\varphi$ map $a$ to $\varphi(a)=ab$ and $b$ to $\varphi(b)=bab$. Then $\varphi(M)$ goes from state 0 to state 1 iff the read letter is $\varphi(a)$. Therefore, $M$ and $\varphi(M)$ are isomorphic. In fact, for the above $M$, we get an isomorphic automaton if and only if exactly one of $\varphi(a)$ and $\varphi(b)$ have an odd number of $b$'s.
In general, for what $M$ and $\varphi$ will be $\varphi(M)$ isomorphic to $M$?
If it makes the question simpler, we can ignore which states are accepting and which is the start state. I am interested in any type of conditions, as I could not find anything on the subject. I am also curious when $\varphi(M)$ will be minimal for the language it recognizes.