Solve for the kernel in a system of Volterra equations

Question

I have a system of $n = 1, \dots, N$ Volterra equations with the same kernel $K$: $$f_n(t) = \int_{0}^{t} K(t, s) g_n(s) \mathrm{d}s\,, \quad t \in [0, 1] \,,$$ where $f_n$ and $g_n$ are known and $K$ is unknown. Further, $\{g_1, \dots, g_N\}$ is a basis for an $N$-dimensional function space. I know that in the case of a single equation there are infinitely many solutions for $K$. I cannot figure out whether this still holds in the $N$-equation case, or if it's now possible that there is no solution? And if there still are solutions, how would I combine them? I was also wondering if there is a literature that studies the unknown kernel case since I did not find any material?