Is every vector bundle connection on this $V$ induced by some connection on $P$?

Question

Let $\pi : P \rightarrow M$ be a $C^\infty$ principal $G$-bundle. Suppose $\pi' : V \rightarrow M $ is an associated vector bundle of $P$, then we know that every connection on $P$ induces a vector bundle connection on $V$. Our question is, Is every vector bundle connection on this $V$ induced by some connection on $P$ ? We are comfortable with the notion of horizontal bundles and connection in terms of smooth horizontal fields. Any help is highly appreciated.