This may be a bit of a silly question, but I think it is useful to get a visual idea of the concept of group extensions. A short exact sequence $1 \to A \overset{i}{\to} B \overset{\pi}{\to} C \to 1$ can be diagrammatically represented as follows (where cosets of $A$ in $B$, represented by horizontal rows, get surjectively mapped to $C$; $A$ is mapped injectively into a horizontal row in $B$):
$B$ is basically the group that is generated using groups $A$ and $C$.
Question 1:
A split extension is defined as an extension $1 \to A \overset{i}{\to} B \overset{\pi}{\to} C \to 1$ with a homomorphism $s: B \to C$ such that going from $B$ to $C$ by $s$ and then back to $B$ by the quotient map of the short exact sequence induces the identity map on $C$, i.e., $\pi \circ s = \mathrm{id}_C$. In this case, $s$ is called a section and it "splits" the exact sequence.
For general groups $A$ and $C$, is $\mathrm{Im}(s)$ (the image of the homomorphism $s: C \to B$) necessarily a vertical column in the above diagram of $B$ or can it just be a set of one representative element from each horizontal row?
Question 2:
A central extension is one where $A$ lies in $Z(B)$. And let's consider only non-split central extensions so as to exclude direct products. In such a case, can we say anything specific about how $\mathrm{Im}(s)$ might look like in the above diagram? Like, certainly $\mathrm{Im}(s)$ can not be a vertical column, unlike in the case of split extensions(?). But can $\mathrm{Im}(s)$ only be a horizontal row in $B$?
P.S: I took the diagram from this video.
I think that your question is very on point, because it highlights the issue with this diagram: the rectangular shape only makes sense for a split extension to begin with.
For a general extension, what makes sense is that you can divide $B$ in $A$-cosets, and that each coset corresponds to an element of $C$. But the rectangular shape says more than that: it implies that there are columns, that is it implies that there is a natural correspondence between certain elements of each coset. Precisely, there is one row which is completely understood: the one in purple, corresponding to the trivial coset $A$ (the one corresponding to the trivial element of $C$). If you draw a rectangle, then in each row (coset) you have an element "in the same column" as some $a\in A$. This means that you actually have an explicit decomposition (at least setwise ) $B\simeq A\times C$.
But to do that basically amounts to choosing a section, which is only possible when the extension is split. So not only columns don't really make sense when the extension is not split, those columns should actually be defined by a section. So to answer your question: yes, a section should land in a column, by definition of what a column should be.
Of course you can also say that for any extension you can always choose a set-theoretic decomposition $B\simeq A\times C$, and draw your rectangle that way. But this means that the diagram basically ignores the algebraic data (or at least partially: the rows still make sense, but the columns don't). This is a pretty bad idea in my opinion. In this case, a section need not correspond to a column because columns are completely arbitrary and don't make algebraic sense.