In this paper, we answer the following question posed by G. Navarro: suppose that S acts on G coprimely, and that A acts on SG so that S and G are both A-invariant. If X is an S-invariant irreducible character of G, then must X extend to AG if and only if X extends to ACG(S), where X is the Glauberman-Isaacs Correspondent for X under the action of S We prove that the answer to this question is yes.
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