  We give a randomness-efficient homomorphism test in the low soundness regime
for functions, $f: G\to \mathbb{U}_t$, from an arbitrary finite group $G$ to
$t\times t$ unitary matrices. We show that if such a function passes a
derandomized Blum--Luby--Rubinfeld (BLR) test (using small-bias sets), then (i)
it correlates with a function arising from a genuine homomorphism, and (ii) it
has a non-trivial Fourier mass on a low-dimensional irreducible representation.
  In the full randomness regime, such a test for matrix-valued functions on
finite groups implicitly appears in the works of Gowers and Hatami [Sbornik:
Mathematics '17], and Moore and Russell [SIAM Journal on Discrete Mathematics
'15]. Thus, our work can be seen as a near-optimal derandomization of their
results. Our key technical contribution is a "degree-2 expander mixing lemma''
that shows that Gowers' $\mathrm{U}^2$ norm can be efficiently estimated by
restricting it to a small-bias subset. Another corollary is a "derandomized''
version of a useful lemma due to Babai, Nikolov, and Pyber [SODA'08].
