  Given two algebraic groups $G$, $H$ over a field $k$, we investigate the
representability of the functor of morphisms (of schemes) $\mathbf{Hom}(G,H)$
and the subfunctor of homomorphisms (of algebraic groups) $\mathbf{Hom}_{\rm
gp}(G,H)$. We show that $\mathbf{Hom}(G,H)$ is represented by a group scheme,
locally of finite type, if the $k$-vector space $\mathcal{O}(G)$ is
finite-dimensional; the converse holds if $H$ is not \'etale. When $G$ is
linearly reductive and $H$ is smooth, we show that $\mathbf{Hom}_{\rm gp}(G,H)$
is represented by a smooth scheme $M$; moreover, every orbit of $H$ acting by
conjugation on $M$ is open.
