  Let $X$ be a finite type $A_n^2$-polyhedron, $n \geq 2$. In this paper we
study the quotient group $\mathcal{E}(X)/\mathcal{E}_*(X)$, where
$\mathcal{E}(X)$ is the group of self-homotopy equivalences of $X$ and
$\mathcal{E}_*(X)$ the subgroup of self-homotopy equivalences inducing the
identity on the homology groups of $X$. We show that not every group can be
realised as $\mathcal{E}(X)$ or $\mathcal{E}(X)/\mathcal{E}_*(X)$ for $X$ an
$A_n^2$-polyhedron, $n\ge 3$, and specific results are obtained for $n=2$.
