$\DeclareMathOperator\Mod{Mod}$Let $S$ be a surface and $P=\{a_1,...,a_n\}$ be a pants decomposition of $S$. Denote by $\Mod(S)$ the mapping class group of $S$. Define the stabilizer of $\Mod(S)$ on $P$ to be $$A=\{f\in \Mod(S), f(a_i)=a_i, i =1,...,n\}.$$ What is $A$? I was once told that it is generated by Dehn twists and hyperelliptic involution, but I cannot find a reference for that. Is it true, and why? Could you give a reference for this?

I saw this on Lemma 3.2 in this paper by U. Wolf, which says that $A$ is generated by Dehn twists and half twists But I didn't understand it yet.