The paper proves that if a normal finite-index subgroup of G is H-accessible or AH-accessible, then G itself is H-accessible or AH-accessible.
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Finite extensions of $\mathcal{H}-$ and $\mathcal{AH}-$accessible groups
The paper proves that if a normal finite-index subgroup of G is H-accessible or AH-accessible, then G itself is H-accessible or AH-accessible.