  We generalize the Plesken-Fabia\'nska $\mathrm{L}_2$-quotient algorithm for
finitely presented groups on two or three generators to allow an arbitrary
number of generators. The main difficulty lies in a constructive description of
the invariant ring of $\mathrm{GL}(2, K)$ on $m$ copies of $\mathrm{SL}(2, K)$
by simultaneous conjugation. By giving this description, we generalize and
simplify some of the known results in invariant theory. An implementation of
the algorithm is available in the computer algebra system Magma.
