Let \(G\) be a group of the form \(G=F\leftthreetimes\langle a\rangle\), where \(a\) is an element of prime order \(p\). The authors investigate the relation between the following conditions: (1) \(a\) acts regularly on \(F\); (2) each element of \(G\setminus F\) has order \(p\); (3) for every \(b\) in \(F\), the subgroup \(\langle a,b\rangle\) is finite; (4) each element of \(F\) is conjugate to \(a\). Various consequences of these properties are studied. For instance, it is shown that a residually finite group with properties (1) and (2) is locally finite.