  For a group $G$ acting on a set $X$, let $\text{End}_G(X)$ be the monoid of
all $G$-equivariant transformations, or $G$-endomorphisms, of $X$, and let
$\text{Aut}_G(X)$ be its group of units. After discussing few basic results in
a general setting, we focus on the case when $G$ and $X$ are both finite in
order to determine the smallest cardinality of a set $W \subseteq
\text{End}_G(X)$ such that $W \cup \text{Aut}_G(X)$ generates
$\text{End}_G(X)$; this is known in semigroup theory as the relative rank of
$\text{End}_G(X)$ modulo $\text{Aut}_G(X)$.
