The higher sharp II: on $M_2^\#$

We establish the descriptive set theoretic representation of the mouse $M_n^{\#}$, which is called $0^{(n+1)\#}$. This part partially deals with the case $n=2$ by proving the many-one equivalence of $M_2^{\#}$ and the theory of $L_{\boldsymbol{\delta}^1_3}[T_3]$ with the higher level analogs of $L$-indiscernibles.