Is the monster group a characteristic quotient of $F_2$?

Let $F_2$ be the free group on two generators, and $M$ the monster group.

It's known that every finite simple group is 2-generated, so let $F_2\rightarrow M$ be a surjection with kernel $N$.

Let $Aut^+(F_2)$ be the subgroup of $Aut(F_2)$ consisting of automorphisms whose image in $Aut(\mathbb{Z}^2)$ has determinant 1.

My questions are:

1. Could $N$ characteristic be $F_2$ (ie, is $N$ invariant under $Aut(F_2)$)?

2. If not, then is $N$ invariant under $Aut^+(F_2)$?

• Your question is unclear because there are zillions of possible $N$. Since it is certainly not characteristic for all $N$ I am guessing you want to know if there eixts such an $N$. Probably not. The images of all primitive elements of $F$ would have to have the same order which seems unlikely. – Derek Holt Aug 21 '15 at 7:45