$(E,p,M)$ is a fiber bundle.
Suppose there exists a local trivialization $LT_1$ of $E$ such that on the overlapping areas the transition can be realized by left action of a topological group $G$ (details of the definition of structure group are omitted here), then we say that $E$ is a $G$-bundle and it has structure group $G$. Also this can induce a principle $G$-bundle $P$.
There is also another local trivialization $LT_2$ and topological group $H$ that satisfy the above and so $E$ is also a $H$-bundle and it has structure group $H$ and there is the induced principle $H$-bundle $Q$.
If $G$ and $H$ are related, e.g. there is homomorphism $H \to G$ or $H \lt G$. In this case, is the principle bundle $Q$ a reduction of structure group of $P$?