Some generalizations of Camina pairs and orders of elements in cosets

In this paper, we investigate certain generalizations of Camina pairs. Let H be a nontrivial proper subgroup of a finite group G. We first show that every nontrivial irreducible complex character of H induces homogeneously to G if and only if for every x∈ G H, the element x is conjugate to $xh$ for all h∈ H. Furthermore we prove that if $xh$ is conjugate to either x or x⁻¹ for all h∈ H and all x∈ G H, then the normal closure N of H in G also satisfies the same condition, and N is nilpotent. Finally, we determine the structure of H under the assumption that for every element x∈ G H of odd order, the coset $xH$ consists entirely of elements of odd order.