In this paper we construct nontrivial pairs of $\frak{S}$-related (i.e. Smith equivalent) real
G-modules for the group
G =
PΣ
L(2,27) and the small groups of order 864 and types 2666, 4666. This and a theorem of K. Pawalowski-R. Solomon together show that Laitinen's conjecture is affirmative for any finite nonsolvable gap group. That is, for a finite nonsolvable gap group
G, there exists a nontrivial $\mathscr{P}$(
G)-matched pair consisting of $\frak{S}$-related real
G-modules if and only if the number of all real conjugacy classes of elements in
G not of prime power order is greater than or equal to 2.
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