Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well.

Let $g\colon M^{2n}\rightarrow M^{2n}$ be a smooth map of period $m>2$ which preserves orientation. Suppose that the cyclic action defined by $g$ is regular and that the normal bundle of the fixed point set $F$ has a $g$-equivariant complex structure. Let $F\pitchfork F$ be the transverse self-intersection of $F$ with itself. If the $g$-signature $\Sign (g,M)$ is a rational integer and $n<\phi (m)$, then there exists a choice of orientations such that $\Sign(g,M)= \Sign F=\Sign(F\pitchfork F)$.