Let $G$ be a finite $n$-generated group and $p$ be a prime not dividing the order $\vert G\vert$; let $C_p$ be the cyclic group of order $p$. According to results by A. Lucchini (Rend. Sem. Mat. Univ. Padova 98 (1997) 67--87 and Arch. Math. 62 (1994) 481--490) the wreath product $C_p^{n-1}\wr G$ is also $n$-generated. Can one write down explicitly a set of $n$ generators of this wreath product (involving generators of $G$)?
Edit: what about the case $n=2$? One would need a 2-generating set for $C_p\wr G\cong \mathbb{F}_pG\rtimes G$ in terms of a 2-generator set of $G$.