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In Permissions API v1, add a static private class _AclObjectIdentity extending AclObjectIdentity with

public List<AclEntry> inherited;

In method lazyLoadForJson, return _AclObjectIdentity and fill the inherited list recursively with ACL entries for all parentObjects:

private void inhertied(AclObjectIdentity objectIdentity, List<AclEntry> aclEntries, Set<AclObjectIdentity> handled) {
  if (objectIdentity == null || handled.contains(objectIdentity)) {
    return aclEntries;
  }
  aclEntries.addAll(objectIdentity.getAclEntries());
  handled.add(objectIdentity);

  if (objectIdentity.getParentObject() != null) {
    inherited(objectIdentity.getParentObject(), aclEntries, handled);
  }
}

Also add objectIdentity.inherited.forEach(entry -> entry.getAclSid().getId());.