  We analyze the problem of estimating a signal from multiple measurements on a
$\mbox{group action channel}$ that linearly transforms a signal by a random
group action followed by a fixed projection and additive Gaussian noise. This
channel is motivated by applications such as multi-reference alignment and
cryo-electron microscopy. We focus on the large noise regime prevalent in these
applications. We give a lower bound on the mean square error (MSE) of any
asymptotically unbiased estimator of the signal's orbit in terms of the
signal's moment tensors, which implies that the MSE is bounded away from 0 when
$N/\sigma^{2d}$ is bounded from above, where $N$ is the number of observations,
$\sigma$ is the noise standard deviation, and $d$ is the so-called
$\mbox{moment order cutoff}$. In contrast, the maximum likelihood estimator is
shown to be consistent if $N /\sigma^{2d}$ diverges.
