Groups for which it is easy to detect graphical regular representations

We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G,S) digraphical regular representation (that is, its automorphism acts regularly on vertex set) or there nontrivial φ such φ(S) = S. show nilpotent p-group, but wreath product Zp wr not DRR-detecting, odd prime p. also if G1 and G2 are groups admit gcd(|G1|, |G2|) 1, then direct x DRR-detecting. Some these results have analogues graphical representations.