Consider a matrix $$ A = \begin{pmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{22} \\ \vdots & & \ddots\\ A_{n1} & A_{n2} & \cdots & A_{nn} \end{pmatrix} \, .$$
Suppose we also have another matrix $B$ with $$B_{ij} = \sum_{kl}c_{ijkl} A_{kl} \, .$$ Is there a well defined way to invert $c_{ijkl}$ so that we can express elements of $A$ in terms of elements of $B$?
The situation you've described is that there are $n^2$ elements ($B_{ij}$), each of which is expressed as a particular linear combination of $n^2$ other elements ($A_{ij}$).
This is more suited to be modelled as an equation of the form $CA=B,$ where $A$ and $B$ are vectors corresponding to the $n^2$ elements of $A_{ij}$ and $B_{ij}$ respectively, and $C$ is the $n^2$ x $n^2$ matrix of coefficients corresponding to your $c_{ijkl}.$
Then a reasonable interpretation of "inverting" $c_{ijkl}$ would be writing $C^{-1}B=A$ when $C$ is invertible.