Given a split extension W ⋊ G, where W is an arbitrary Coxeter group and G a group of automorphisms of the Coxeter graph of W, we determine all involutions in W ⋊ G whose centralizers are of finite index. Our result has applications to many problems such as the isomorphism problem for general Coxeter groups. In the course of the proof, some properties of certain special elements and of fixed-point subgroups of graph automorphisms in Coxeter groups which are of independent interest are given.
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