You need to establish what the contractual arrangements are. It’s likely to be either:
1. There is a direct contract for the work between you and the ultimate customer, and this other company is acting as your agent in making the arrangements; or
2. There is no direct contact between you and the ultimate customer – instead you are contracted to supply services to this other company, and they are contracted to supply services on to the ultimate customer.
From your question, I’ll assume it’s the first of these. If it’s the second one, let me know and I’ll re-work it. I’m also going to assume that you’re not involved in the construction industry, as there are some different rules there. Again, let me know if it is construction-related.
The other company is collecting the £78 on your behalf. The £13 VAT element of this is for you to pay to HMRC on your VAT return, not for the other company to deal with. The £13 (incl VAT) fee deduction that the other company makes is an expense to you, so you’re entitled to reclaim the VAT element of this (£2.17) from HMRC – you’ll need a VAT receipt or receipted VAT invoice from them to support this, in the same way as any other business expense.
If this is the case, the other company should pay you £65.00 – the amount that they’ve collected from the ultimate customer, less their fee. You will need to pay HMRC the £13.00 due on the ultimate customer’s payment, less the £2.17 recoverable on the other company’s fee. That leaves you £54.17, which is the £65.00 from other company, less the £13.00 VAT you pay to HMRC, plus the £2.17 VAT on the fee, that you reclaim from HMRC.