The purpose of this paper is to give positive answers to some questions which are related to Fox, Rhodes, Gottlieb-Fox, and Gottlieb-Rhodes groups. Firstly, we show that for a compactly generated Hausdorff based G-space (X, x 0 , G) with free and properly discontinuous G-action, if (X, x 0 , G) is homotopically n-equivariant, then the n-th Gottlieb-Rhodes group Gσ n (X, x 0 , G) is isomorphic to the n-th Gottlieb-Fox group Gτ n (X/G, p(x 0 )). Secondly, we prove that every short exact sequence of groups is n-Rhodes-Fox realizable for any positive integer n. Finally, we present some positive answers to restricted realization problems for Gottlieb-Fox groups and Gottlieb-Rhodes groups.