Consider an equation of the form AX=A where A is $m \times n$ $m<n$ matrix and X is an unknown $n \times n$ invertible matrix all matrices are over $\mathbb{F}_2$.
Can one characterize A when is there a non-identity solution to X and when $X=I_n$ is the only solution?
As I have stated in a comment above, $X = I$ will be the only solution if and only if $A$ has full column-rank.
Because $\Bbb F_2$ is a field, the standard linear algebraic approaches apply.
Note that $AX = A$ if and only if $A(X - I) = 0$, and $A(X - I) = 0$ if and only if the column space of $X - I$ is contained in the null space of $A$. If this only occurs when $X - I = 0$, then the null space of $A$ is trivial, which is to say that $A$ has full column-rank.