Is solving systems of equations modulo $k$ in $\mathsf{coMod}_k\mathsf L$ for $k$ composite?
I'm happy to say that I think we can answer this question in the affirmative: that is, deciding whether a linear congruence is feasible modulo k is coModkL-complete.

We can actually reduce this problem to the special case of prime powers. One may show that: 


  Normal Form. The class coModkL consists of langauges L of the form L&nbsp;=&nbsp;Lp1&nbsp;&cap;&nbsp;Lp2&nbsp;&cap;&nbsp;...&nbsp;&cap;&nbsp;Lpr&nbsp;, where Lpj&nbsp;&isin;&nbsp;coModpj&nbsp;L and where pj ranges over the prime factors of k.


By the Remainder Theorem, any solution to a system of equations modulo each of the prime powers $p_{\!j}^{e_j}$ dividing k gives rise to a solution to the same system, mod k.
So if solving systems of linear equations over $p_{\!j}^{t_j}$ is contained in coModpj&nbsp;L, it follows that solving systems of equations mod k is contained in coModkL.

There's a standard algorithm, described by McKenzie and Cook for reducing linear congruences modulo a prime power to constructing a spanning set for its nullspace (namely, for Ax&nbsp;=&nbsp;y over a given ring, construct a basis for the nullspace of [&nbsp;A&nbsp;|&nbsp;y&nbsp;] and see if any solutions exist with a final coefficient of &minus;1); and subsequently for reducing the construction of nullspaces modulo prime powers to constructing nullspaces modulo primes, and matrix multiplication modulo prime powers. Both of the latter tasks are problems which are feasible for coModkL, provided that you can construct the matrices involved.

It turns out that the matrices involved in the reduction of McKenzie and Cook can themselves be computed by matrix multiplication, and (crucially) division by a constant factor. Fortunately, for prime powers, the coefficients of the matrices involved can be computed on the work-tape using an oracle for coModpL-machines; and the division by a constant can be performed in NC1, which is again feasible in coModpL. So it turns out that the whole problem is ultimately doable in coModkL.

For complete details, see [arxiv:1202.3949].