Let $Z$ be a group and consider the symmetric group $S_n$ on $n$ elements. Assume we have an action $\phi$ of $S_n$ on $Z$ by automorphisms. Consider the group $K$ which is a direct product of $n$ copies of $Z$.
Consider now the action $\psi$ of an element $\sigma$ of $S_n$ on $K$ which sends a $n$-tuple $(z_1,z_2,...,z_n) \in K, z_i \in Z$ to $(f(z_1),f(z_2),...,f(z_n))$ where $f(z_i)=\phi_\sigma(z_{\sigma(i)})$ if $\sigma(i) \neq i$, and $f(z_i)=z_i$ if $\sigma(i)=i$.
Consider finally the semidirect product of $K$ by $S_n$, with the action $\psi$. My question is: is it a wreath product of $Z$ by $S_n$ ? Or is it a more general construction ? I've tried considering building explicit isomorphisms to $Z \wr S_n$, but the action of $\phi$ is problematic.