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2 deleted superfluous "relatively free"

Does there exist a variety of groups $\mathfrak{V}$ such that the relatively $\mathfrak{V}$-free group of rank 2 is finite, but the relatively $\mathfrak{V}$-group of rank 3 is infinite?

(In other varieties of algebras this can occur; for example, in the variety of all lattices, the free lattice of rank 2 is finite, but the relatively free lattice of rank 3 is infinite.)

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# Varieties of groups with infinite relatively free group of rank 2 finite, infinite in rank 3

Does there exist a variety of groups $\mathfrak{V}$ such that the relatively $\mathfrak{V}$-free group of rank 2 is finite, but the relatively $\mathfrak{V}$-group of rank 3 is infinite?

(In other varieties of algebras this can occur; for example, in the variety of all lattices, the free lattice of rank 2 is finite, but the relatively free lattice of rank 3 is infinite.)