  Assuming that every set is constructible, we find a $\Pi^1_1$ maximal
cofinitary group of permutations of $\mathbb N$ which is indestructible by
Cohen forcing. Thus we show that the existence of such groups is consistent
with arbitrarily large continuum. Our method also gives a new proof, inspired
by the forcing method, of Kastermans' result that there exists a $\Pi^1_1$
maximal cofinitary group in $L$.
