Let $H,G,K$ be three topological groups, we say that $G$ is an extension of $K$ by $H$ if the following short sequence $$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0$$ is exact. (If $H$ is a subgroup of $G$ this is equivalent to $K\cong G/H$)

Now, assume that $H,G,K$ are compact abelian topological groups, therefore, by the structure theorem for compact abelian Lie groups, we have that $K$ is a Lie group if and only if it is isomorphic to $\mathbb{T}^n\times C_k$ where $\mathbb{T}$ is the circle group ($S^1$ or $\mathbb{R}/\mathbb{Z}$), $C_k$ is a finite abelian group and $n\in\mathbb{N}$.

It is well known that every extension of a Lie group by a Lie group is again a Lie group. In particular, every extension of a Lie group by a finite group is a Lie group.

My question is: whether every extension of a compact abelian Lie group by a pro-finite group is a subgroup of $\mathbb{T}^n\times D$ where $D$ is a profinite group for some $n\in\mathbb{N}$.

I am most interested in the case where $H$ is a direct product of finite groups.

Any proof / reference is appreciated.

Edit: I also assume that the groups are Hausdorff. Also every morphism is both topological and algebraic.