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Let p be a prime number. A positive integer m is said to be a p ′ -number provided p ∤ m . Let G be a finite group, and let H be a subgroup of G. We say that H is weakly s p -permutable in G if G has a subnormal subgroup K such that G = HK, H sG ≤ K and | H ∩ K : H sG | is a p ′ -number, where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We study the structure of a finite group G of even order all of whose fourth maximal subgroups are weakly s 2 -permutable in G. [ABSTRACT FROM AUTHOR]