  We report on a study of the Gamow-Teller matrix element contributing to
${}^6$He $\beta$-decay with similarity renormalization group (SRG) versions of
momentum- and configuration-space two-nucleon interactions. These interactions
are derived from two different formulations of chiral effective field theory
($\chi$EFT) -- without and with the explicit inclusion of $\Delta$-isobars. We
consider evolution parameters $\Lambda_{\rm SRG}$ in the range between 1.2 and
2.0 fm$^{-1}$ and, for the $\Delta$-less case, also the unevolved (bare)
interaction. The axial current contains one- and two-body terms, consistently
derived at tree level (no loops) in the two distinct $\chi$EFT formulations we
have adopted here. The ${}^6$He and ${}^6$Li ground-state wave functions are
obtained from hyperspherical-harmonics (HH) solutions of the nuclear many-body
problem. In $A\,$=$\,6$ systems, the HH method is limited at present to treat
only two-body interactions and non-SRG evolved currents. Our results exhibit a
significant dependence on $\Lambda_{\text{SRG}}$ of the contributions
associated with two-body currents, suggesting that a consistent SRG-evolution
of these is needed in order to obtain reliable estimates. We also show that the
contributions from one-pion-exchange currents depend strongly on the model
(chiral) interactions and on the momentum- or configuration-space cutoffs used
to regularize them. These results might prove helpful in clarifying the origin
of the sign difference recently found in No-Core-Shell-Model and Quantum Monte
Carlo calculations of the ${}^6$He Gamow-Teller matrix element.
