Inner automorphism group of wreath product of cyclic group of prime power order and group of order p

Definition
Let $$p$$ be a prime number. The inner automorphism group of wreath product of cyclic group of prime power order and group of order p is a group obtained as follows: it is the inner automorphism group of the wreath product of cyclic group of prime power order and group of order p, where the acting group acts regularly: it acts by cyclic permutation of $$p$$ coordinates.

In other words, if $$C_p$$ denotes the cyclic group of order $$p$$, and $$C_{p^k}$$ is a cyclic group of order $$p^k$$, and $$G = C_{p^k} \wr C_p = (C_{p^k} \times \dots \times C_{p^k}) \rtimes C_p$$, then the group we want is $$H = G/Z(G)$$. It turns out that $$Z(G)$$ is precisely the diagonal subgroup of $$C_{p^k} \times \dots \times C_{p^k}$$.

$$H$$ is a $$p$$-group of order $$p^{k(p-1) + 1}$$.

A particular case of interest is the inner automorphism group of wreath product of groups of order p.