  Let $f : X \rightarrow Y$ be a generically smooth nonconstant morphism
between irreducible projective curves, defined over an algebraically closed
field, which is \'etale on an open subset of $Y$ that contains both the
singular locus of $Y$ and the image, in $Y$, of the singular locus of $X$. We
prove that the following statements are equivalent: \begin{enumerate} \item The
homomorphism of \'etale fundamental groups $$f_* : \pi_1^{\rm et}(X)
\rightarrow\pi_1^{\rm et}(Y)$$ induced by $f$ is surjective.
  \item There is no nontrivial \'etale covering $\phi : Y' \rightarrow Y$
admitting a morphism $q: X \rightarrow Y'$ such that $\phi\circ q = f$.
  \item The fiber product $X\times_Y X$ is connected.
  \item $\dim H^0(X, f^*f_* {\mathcal O}_X)= 1$.
  \item ${\mathcal O}_Y \subset f_*{\mathcal O}_X$ is the maximal semistable
subsheaf.
  \item The pullback $f^*E$ of every stable sheaf $E$ on $Y$ is also stable.
\end{enumerate}
