Subgroups of finite soluble groups inducing the same permutation character 
Norberto Gavioli
Trans. Amer. Math. Soc. 349 (1997), 2969-2980. 

Abstract. In this paper there are found necessary and sufficient conditions 
that a pair of solvable finite groups, say G and K, must satisfy for the 
existence of a solvable finite group L containing two isomorphic copies of G 
and H inducing the same permutation character. Also a construction of L is
given as an iterated wreath product, with respect to their actions on their 
natural modules, of finite one-dimensional affine groups.