Free groups as end homogeneity groups of 3-manifolds

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 homogeneity of isomorphic $F$. The all self-homeomorphisms that extend homeomorphisms entire $3$-manifold. This extends earlier result constructs, for each abelian $G$, $M_G$ $G$. method used in proof our main also shows if $G$ Cayley graph $\mathbb{R}^3$ such automorphisms have certain nice extension properties, then there