  For every finitely generated free group $F$, we construct an irreducible open
$3$-manifold $M_F$ whose end set is homeomorphic to a Cantor set, and with the
end homogeneity group of $M_F$ isomorphic to $F$. The end homogeneity group is
the group of all self-homeomorphisms of the end set that extend to
homeomorphisms of the entire $3$-manifold. This extends an earlier result that
constructs, for each finitely generated abelian group $G$, an irreducible open
$3$-manifold $M_G$ with end homogeneity group $G$. The method used in the proof
of our main result also shows that if $G$ is a group with a Cayley graph in
$\mathbb{R}^3$ such that the graph automorphisms have certain nice extension
properties, then there is an irreducible open $3$-manifold $M_G$ with end
homogeneity group $G$.
