Given a transformation group \((X,G)\), write \(\sigma_n(X,x_0,G)\) and \(G\sigma_n(X,x_0,G)\), where \(x_0\) is a base point of \(X\), for its \(n\)-th Rhodes (cf. \textit{F. Rhodes} [Can. J. Math. 21, 1123--1136 (1969; Zbl 0183.28102)]) and Gottlieb-Rhodes groups, respectively. If \(G=E\), the trivial group, then this yields its \(n\)-th Fox (cf. \textit{R. H. Fox} [Ann. Math. (2) 49, 471--510 (1948; Zbl 0038.36602)]) and Gottlieb-Fox groups, respectively. The purpose of this paper is to give some positive answers to the following problems: (i) when are the Gottlieb-Rhodes groups of a space \(X\) with free and properly discontinuous \(G\)-action isomorphic to the Gottlieb-Fox groups of its orbit space \(X/G\)? (ii) a realization problem for Rhodes groups; (iii) some restricted realization problems for Fox, Rhodes, Gottlieb-Fox and Gottlieb-Rhodes groups.