This is finite state automaton for Grigorchuk group. I have never studied automaton formally, so I wanna check is it fine the way I am doing it. Here $\epsilon$ change the first entry on string from 0 to 1 and vica versa.
If I calculate $d(10011)$ that means I put $10011$ in d circle, $1$ inside circle means identity automorphism i.e. $1_e$ , so it remains $10011$ then as first digit is $1$ so it goes to b, so now will b act $1_e$ on $0011$ i.e. it goes to $(1, b(0011))$ and then under $1_e$ 0011 remains 0011 and then detecting 0 it goes to $\epsilon$ which acts on 011 and it becomes 10111 and then goes to final state. I think I am correct with this understanding of automaton working,
but now the problem is if I calculate $b(10)$ it goes to $(1,c(0))$, now reading $0$ c sends it to a, which has to apply $\epsilon$ to $\phi$ (empty string), what does this mean. What is then $b(0,1)$, it didn't reach final state.