  Motivated in part by the study of the stable homology of automorphism groups
of free groups, we consider cohomological calculations in the category
$\mathcal{F}(\textbf{gr})$ of functors from finitely generated free groups to
abelian groups.In particular, we compute the groups
$Ext^*\_{\mathcal{F}(\textbf{gr})}(T^n \circ \mathfrak{a}, T^m \circ
\mathfrak{a})$ where $\mathfrak{a}$ is the abelianization functor and $T^n$ is
the n-th tensor power functor for abelian groups. These groups are shown to be
non-zero if and only if $*=m-n \geq 0$ and
$Ext^{m-n}\_{\mathcal{F}(\textbf{gr})}(T^n \circ \mathfrak{a}, T^m \circ
\mathfrak{a})=\mathbb{Z}[Surj(m,n)]$ where $Surj(m,n)$ is the set of
surjections from a set having $m$ elements to a set having $n$ elements. We
make explicit the action of symmetric groups on these groups and the Yoneda and
external products. We deduce from these computations those of rational
Ext-groups for functors of the form $F \circ \mathfrak{a}$ where $F$ is a
symmetric or an exterior power functor. Combining these computations with a
recent result of Djament we obtain explicit computations of stable homology of
automorphism groups of free groups with coefficients given by particular
contravariant functors.
