In this paper a thoroughgoing study of hypergroupoids considered by the author in a previous work [Matematiche 52, No. 2, 271-295 (1997; Zbl 0941.20071)] is done. Let \(G\) be a group, \(\lambda\in G\) and \(\cdot\colon G\times M\) be an action of \(G\) on \(M\). The aim of this paper is to investigate the hyperoperation defined on \(M\) by: \(a\cdot b=b\cdot a=\{\lambda a,\lambda b\}\) for every \(a\), \(b\) in \(M\). Necessary and sufficient conditions in order that \((M,\cdot)\) be matroidal (i.e., it satisfies the exchange condition) are given. The dimension of matroidal hypergroupoids (in the sense of G. Tallini) is calculated and properties of morphisms of matroidal hypergroupoids are analyzed. The cardinal of the automorphism group of the hypergroupoid \((G,\cdot)\), \(G\) being finite, is found.