The author studies finite groups \(G\) generated by a class \(D\) of conjugate involutions which contain a subgroup \(S\) isomorphic to the symmetric group \(S_ n\) such that \(S\cap D=:\Delta\) corresponds to the class of transpositions of \(S_ n\) and \(S\) acts transitively on \(D-\Delta\). Theorem 1: If for all \(d_ 1\neq d_ 2\) from \(D-\Delta\) we have \(C_{\Delta}(d_ 1)\neq C_{\Delta}(d_ 2)\), then \(D\) is a class of 3- transpositions (i.e. \(| a\cdot b| \leq 3\) for all \(a,b\in D)\). Theorem 2: If for all \(d\in D-\Delta\) we have \(C_{\Delta}(d)=\emptyset\), then \(n\leq 4\), or \(G=NS\) for the maximal solvable normal subgroup \(N\) of \(G\), or \(n=6\), \(G\cong S_ 7\), \(D\) the class of type \((2)^ 3\). The following further example from the paper is not covered by these theorems: \(G\cong A_ 8\), \(S\cong S_ 6\) the normalizer of a two-element set, \(D\) the class of type \((2)^ 4\). -- In Theorem 1 the elements \(d\) of \(D-\Delta\) can be distinguished by the orbits of \(\) on \(\{1,\ldots,n\}\). For \((12)\in \Delta\) and arbitrary \(x\in D-\Delta\), about 30 possibilities for the orbit structures of the pair \(x,y=(12)^ x\) are discussed to obtain \(| (12)\cdot x| \leq 3\).