Hi,
HMRC”s detailed guidance is in notice 700/2 https://www.gov.uk/guidance/group-and-divisional-registration-vat-notice-7002. But, assuming that all three members of the group are UK based, you need to consider the group as if it was a single organisation.
No VAT is charged on the management charge because the supply is “internal”, within the group. The ability of the group to recover VAT charged by “external” suppliers is dependent upon whether it relates to taxable supplies made by the group to “external” customers.
If the only supplies made by the group to the “external” customers are VAT exempt, the group won’t be entitled to input tax recovery.