What is a company allowed to do with investors funds?

Your logic doesn't match exactly, because the stock price going up isn't going to happen (because, why should it?)

But let's simplify it a bit:

Imagine there are two companies. For the sake of simplicity, Company A has £1m, Company B has nothing (or nearly nothing, or maybe just £1).

Company A invests these £1m in Company B. For the sake if this, B creates new shares. As B's owners' part of the company is diluted, they have to agree to this, but let's assume they do.

Then B has £1m of cash and invests them in A. In order to make this happen, A creates new shares (the owners agreeing), and is now worth £2m.

At this point, the following happens:

A is worth £2m. It has £1m of cash and £1m of shares in B. A belongs to its original owners (50%) and to B (50%).

B is worth £1m. All this consists of £1m of shares in A (and £1 in cash, but let's ignore that). B belongs to A (£1m) and to its original owner (£1, but let's ignore that as well).

Now what would be the purpose of doing so?

A modified form of the @glglgl example:

1. Company B issues new shares (thus diluting existing ownership).
2. Company A can buy them for £1m cash, thus injecting new money into Company B.
3. Now Company B has a £1m asset of type "cash".
4. For whatever foolish reason, Company B uses that £1m asset to buy £1m of share of Company A stock.
5. Thus, Company B no longer has a £1m asset of type "cash", but now has a £1m asset of type "shares" in Company A.

As you can see, there's no magical money expansion. B has £1m more in assets than it did at the beginning of the exercise.