Suppose $P$ a principal bundle over connected manifold $B$ with correspondent Lie group $G=SU(2)$, and $A$ a connection on $P$.
We say a map $\sigma \in Aut(P)$ a stabilizer of $A$ if $\sigma^*A=A$.There is a theorem say that for connected manifold $X$ the restriction map to a fiber $P_x$ induced isomorphism between stabilizer of connection and the subgroup of $G$ which centralizes the holonomy group w.r.t. to connection $A$ at point $x$, thus it's make sense to say a connection a $H-$connection, which means the stabilizer is correspondent to a subgroup $H$ of $G$ centralizes the holonomy group.
My question is, suppose $A$ a $U(1)\times \mathbb{Z}/{2\mathbb{Z}}-$connection, how to show that $A$ is a induced connection on a $S^1\cong U(1)$ bundle? This question is exactly the corollary 4.3.5 of John Morgan's book Gauge Theory and Topology of Four-manifolds but I think the proof they give is quite unclear.
Thanks in advance.