Are group theoretic splittings of Lie groups automatically differentiable?

Question

Suppose that $G$ is a Lie group, and that $N$ is a normal Lie subgroup of $G$. Then $G / N$ is also a Lie group.

If $0 \to N \to G \to G/N \to 0$ splits as groups (i.e. $G$ is a semidirect product of $N$ and $G/N$ as abstract groups, i.e. there is some not necessarily smooth section $G/N$ to $G$), with splitting $\gamma : G / N \to G$, then does $G$ also split as Lie groups, i.e. the image of $\gamma$ is a Lie subgroup of $G$?

A priori I see no reason for $\gamma$ to be smooth, also there is the issue that the smooth image of a Lie group may not be a Lie group.

If the image of the splitting $\gamma$ was a manifold, then the restriction $p : G \to G / N$ of the projection to it is smooth bijective homomorphism between Lie groups, hence a diffeomorphism. So in this case $\gamma$ would be smooth.

So the question is really: Is $\gamma(G/N)$ a manifold? (I suspect no, but is there a good example?)

Answer 1

Suppose that $N$ is commutative, the splitting extensions are classified by $H^1(G,N)$ which represents here the discrete or the continuous cohomology. If $N$ is $Q$ and $R$ the discrete cohomology and the Lie cohomology do not coincide generally. You can see for example this paper of Milnor, where the (co) homology relatively to different coefficients are discussed.

http://www.maths.ed.ac.uk/~aar/papers/milnor7.pdf