Corollary 4

Let $(E, p)$ be a connected covering of B and let $x$ be a point of the fibre $E_b$. In order that E be a Galois covering of B, it is necessary and sufficient that $p_*(\pi_1(E, x))$ be a distinguished subgroup of $\pi_1(B, b)$. The group Aut$_B(E)$ is then isomorphic to the quotient group $\pi_1(B, b)/p_*(\pi_1(E, x))$.

If $p_*(\pi_1(E, x))$ is a distinguished subgroup of $\pi_1(B, b)$, the group Aut$_B(E)$ operates transitively on the fibre $E_b$ by Corollary 3, so that E is a Galois covering of B (I, p. 102, th. 2). The converse is assertion d) of Theorem 1 (III, p. 305). The last assertion follows from Corollary 3.