  Clonoids are sets of finitary functions from an algebra $\mathbb{A}$ to an
algebra $\mathbb{B}$ that are closed under composition with term functions of
$\mathbb{A}$ on the domain side and with term functions of $\mathbb{B}$ on the
codomain side. For $\mathbb{A},\mathbb{B}$ (polynomially equivalent to) finite
modules we show: If $\mathbb{A},\mathbb{B}$ have coprime order and the
congruence lattice of $\mathbb{A}$ is distributive, then there are only
finitely many clonoids from $\mathbb{A}$ to $\mathbb{B}$. This is proved by
establishing for every natural number $k$ a particular linear equation that all
$k$-ary functions from $\mathbb{A}$ to $\mathbb{B}$ satisfy. Else if
$\mathbb{A},\mathbb{B}$ do not have coprime order, then there exist infinite
ascending chains of clonoids from $\mathbb{A}$ to $\mathbb{B}$ ordered by
inclusion. Consequently any extension of $\mathbb{A}$ by $\mathbb{B}$ has
countably infinitely many $2$-nilpotent expansions up to term equivalence.
