Following Quillen's programme, one can read off a lot of information on the cohomology of a finite group $G$ by studying the restriction homomorphism from this cohomology to the cohomology of all maximal elementary abelian subgroups of $G$. This leads to a natural question on how much information on $\cal A$-generators of $H^*(G)$ one can read off from using the restriction homomorphism, where $\cal A$ denotes the steenrod algebra. In this paper, we show that the restriction homomorphism gives, in some sense, very little information on $\cal A$-generators of $H^*(G)$ at least in the important three cases, where G is either the symmetric group, the alternating group, or a certain type of iterated wreath products.