  We study phenomenologically inclusive spectra of the $^6$Li($\pi^-$, $K^+$)
reaction at 1.2 GeV/$c$ within a distorted-wave impulse approximation with the
optimal Fermi-averaging $\pi^- p \to K^+ \Sigma^-$ $t$ matrix. We attempt to
clarify the property of a $\Sigma$-nucleus potential for $\Sigma^-$-$^5$He by
comparing the calculated spectra with the data of the J-PARC E10 experiment.
The result shows that the repulsive and absorptive components of the
$\Sigma^-$-$^5$He potential provide the ability to explain the data of the
continuum spectra in $\Sigma$ and $\Lambda$ regions; the strengths of $V_\Sigma
\simeq$ +30 MeV and $W_\Sigma \simeq$ $-$26 MeV are favored within the
Woods-Saxon potential, consistent with analyses for heavier nuclei. Effects of
the size and potential range for $\Sigma^-$-$^5$He in the neutron excess of
$(N-Z)/(N+Z)=$ 0.2 are also discussed.
