On Regulation 63 we can find the rules related to the use of technical and financial capacity from third parties to justify the participation of an economic operator on a public procurement procedure. It transposes Article 63 of Directive 2014/24/EU into England, Wales and Northern Ireland. In general, the reliance is possible irrespective of the nature of the legal links between the economic operator taking part in the procedure, but this general rule is subject to a number of exceptions or specific rules.
Regulation 63 contains specific rules for different scenarios. So, if an economic operator wants to rely on the relevant experience of a third party, it can do so if the third party will undertake the tasks for which said experience is required (paragraph 2). A similar requirement is imposed in situations where the economic operator is relying on capacity (paragraph 3), for example manpower. In this scenario, there must be a clear commitment of the capacity that is being requested from the third party.
Financial capacity also has its own specific rules in paragraphs 5 which allows the contracting authority to impose a joint liability for the performance of the contract, something that should give pause for thought to any third party entity willing to "lend" its financial capacity in this setting.
Paragraph 6 deals with group relationships as they are conceived under Regulation19(3). As such, consortia and groupings of economic operators may rely on the capacity of members of the group or third parties and the same rules of paragraphs 1 to 5 are applicable to them as they would to a single economic operator.
As far as I can tell there are no special dispensations for group relationships, ie when the economic operator detains/controls a third party upon which capacity it relies upon. It will be regulated as any other situation of third party reliance.