Homotopy types of orbit spaces and their self-equivalences for the periodic groups Z/a rtimes (Z/b x T^star_n) and Z/a rtimes (Z/b x O^star_n)

Let G be a finite group given in one of the forms listed in the title with period 2d and X(n) an n-dimensional CW-complex with the homotopy type of an n-sphere. We study the automorphism group Aut(G) to compute the number of distinct homotopy types of orbit spaces X(2dn-1)/\mu with respect to free and cellular G-actions \mu on all CW-complexes X(2dn-1). At the end, the groups E(X(2dn-1)/\mu) of self homotopy equivalences of orbit spaces X(2dn-1)/\mu associated with free and cellular G-actions \mu on X(2dn-1) are determined.