Return to Revisions

The answer is no, also for finite groups. Consider the finite group of order $16$ $$G := \langle a,b,x \mid a^4 = b^2 = x^2 = 1, \, ab = ba, \, bx = xb, \, xax^{-1} = ab \rangle,$$ whose GAP label is $G(16, \, 3)$, and take the two normal subgroups $$H_1 = \langle a, \, b \rangle \simeq \mathbb{Z}_4 \times \mathbb{Z}_2, \quad H_2 = \langle a^2, \, b, \, x \rangle \simeq \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2.$$ They are not isomorphic as abstract groups, so no automorphism of $G$ can send $H_1$ to $H_2$. However, we have $$G/H_1 \simeq G/H_2 \simeq \mathbb{Z}_2.$$