My 2¢... Not an expert, but making educated guesses based on observations thus far. Addressing a few specific queries/comments raised in the original post.

1) Does it make sense for a GEO3 officer to be given the role of a HOD?
Sure, if the school has a need and officer has been identified as capable enough to take on the duties. I would want to clarify if the officer is meant to be acting HOD or to officially take on the appointment, though - there is a difference (to elaborate later).

2) Officer has been stuck on GEO3 scale for 4 years.
This is interesting - I feel that's pretty long for an officer with HOD-level CEP. This is just a guess, though. I'm not high up enough to know the norms.

3) Will the extra responsibilities of a HOD necessarily lead to a quick promotion from GEO3 to GEO5 perhaps?
I think it's quite probable. To be formally appointed as HOD, you will need to have a minimum CEP - that should effectively increase the speed of promotion. If officer is being appointed acting HOD with the intent of officially taking on the role later, the school might be planning to raise the CEP to the prerequisite level before sending for interview. This will also lead to faster promotion.

4) Otherwise, whats the point of having the extra responsibilities if the payscale is going to be the same as fellow peers/or even lower?
At the very minimum, a GEO3 doing HOD work should be appraised very favourably as compared against other GEO3s. Almost confirmed B grade (possibly A, though that's really super rare). Though CEP is assigned/raised separately from performance grade, there will generally be a correlation and I would expect a corresponding increase in CEP/promotion speed to follow, if the job is done well.

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This is of course just the pragmatic career progression aspect of things. There's lots of other stuff to consider (officer's temperament, department dynamics, etc.) but I'm sure they can assess that on their own.

At the end of the day, in their shoes, I'd like some clarity on whether I am to officially take up the role, or if it's only an internal appointment (and if so, whether an official appointment is to be expected). I've heard some horror stories where the acting HOD is never formally appointed and someone else is eventually brought in to take up the role. Then again, in those cases the officer probably did not perform to the school's expectations.

All the best.