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.