If $V\,\to \,X$ is a vector bundle of fiber dimension $k$ and $Y\,\to \,X$ is a finite sheeted covering map of degree $d$, the implications for the Euler class $e(V)$ in ${{H}^{k}}(X)$ of $V$ implied by the existence of an embedding $Y\,\to \,V$ lifting the covering map are explored. In particular it is proved that $d{{d}^{\prime }}\text{e(V)}\text{=}\text{0}$ where ${{d}^{\prime }}$ is a certain divisor of $d\,-\,1$, and often ${{d}^{\prime }}=1$.