On F-hypercentral modules and character clusters

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Let F be a saturated formation of soluble Lie algebras over a field F of characteristic p > 0 and let Fₚ denote the field of p elements. Let (L,[p]) be a restricted Lie algebra over F with z [p]=0 for all z in the centre of L. Let S ∈ F, S≠ 0 be a subnormal subalgebra of L. Let V, W be L-modules. Suppose that the character cluster of W is contained in the set of Fₚ-linear combinations of the characters in the character cluster of V. Suppose that V, regarded as S-module, is F-hypercentral. Then W, regarded as S-module, is also F-hypercentral.