The Cayley graph of a subgroup of the Burnside group $B_0(2,5)$

Let $B_0(2,5)=\langle a_1,a_2\rangle$ be the largest two-generator Burnside group of exponent five. It has the order $5^{34}$. We define an automorphism $\varphi $ under which every generator is mapped into another generator. Let $C_{B_0(2,5)}(\varphi)$ be the centralizer of $\varphi$ in $B_0(2,5)$. It is known that $|C_{B_0(2,5)}(\varphi)|=5^{17}$. We have calculated the growth function of this group relative to the minimal generating set $X$. As a result, the diameter and the average diameter of $C_{B_0(2,5)}(\varphi)$ are computed: $D_X(C)=33$, $\overline D_X(C)\approx26{,}1$.