re:
this question, am I losing my mind or isn't it clear that permutations in the symmetric group $S_g$ acting on $\{a_i\}$ and $\{b_i\}$ simultaneously induce automorphisms of $$\Gamma_g=\langle a_1,b_1,\cdots,a_g,b_g:[a_1,b_1],\cdots,[a_g,b_g]\rangle?$$