  In [11] Sklinos proved that any uncountable free group is not
$\aleph_1$-homogenenous. This was later generalized by Belegradek in [1] to
torsion-free residually finite relatively free groups, leaving open whether the
assumption of residual finiteness was necessary. In this paper we use methods
arising from the classical analysis of relatively free groups in infinitary
logic to answer Belegradek's question in the negative. Our methods are general
and they also applications in varieties with torsion, for example we show that
if $V$ contains a non-solvable group, then any uncountable $V$-free group is
not $\aleph_1$-homogenenous.
