= Solution
For a small peculiar line-of-sight velocity $v_\parallel$ within the group, cosmological and Doppler factors multiply:
$$
1+z_{\rm obs}=(1+z_g)(1+z_{\rm pec}),\qquad z_{\rm pec}\simeq v_\parallel/c.
$$
At fixed group <redshift> this gives $\delta z=(1+z_g)v_\parallel/c$. Therefore
$$
\boxed{\sigma_v=\frac{c\sigma_z}{1+z_g}=99.9\ \mathrm{km\,s^{-1}}}.
$$
Using $c\sigma_z$ directly would give a detector-frame <redshift> interval without removing the cosmological stretch. This is the one-dimensional line-of-sight <velocity dispersion> requested, not the three-dimensional RMS speed. <Redshift> measurement errors or a large unbound line-of-sight extent would require additional corrections; neither is supplied here.
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