In his recent post Bartek demonstrated an example of two equivariant sets with atoms that are related by a finitely supported bijection, but not by any equivariant bijection. We show that if the sets are orbit-finite then this situation is impossible.

**Proposition.** Let and be two orbit-finite equivariant sets with atoms. If there exists a finitely supported bijection then there exists an equivariant bijection .

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