= Solution
For the <fundamental representation> and the <two-index symmetric representation>, the <Dynkin indices> are $1$ and $5$. Therefore the specified discrete <gauge anomaly> condition for $\mathbb Z_4$ is
$$
q_1+5q_2\equiv q_1+q_2\equiv0\pmod4.
$$
All ordered solutions are
$$
\boxed{(q_1,q_2)\in\{(0,0),(1,3),(2,2),(3,1)\}\pmod4.}
$$
Only $(1,3)$ and $(3,1)$ give a faithful $\mathbb Z_4$ action on this pair of <Weyl fermions>; faithfulness was not imposed. As in <2/e/i/solution>, cancellation here concerns the stipulated discrete <gauge anomaly>, not cancellation of the pure cubic <SU(3)> <gauge anomaly>. The integer normalization matters: one must not replace $c$ by a half-integer trace index inside an unchanged modular congruence.
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