I'll give you a bit more help!
Of the 6 cabinet ministers that were Home Secretary in that period, 3 were not present in the cabinet formed by Gordon Brown listed in the site below
https://en.wikipedia.org/wiki/Brown_ministry
of the three remaining, two were not on Blair's cabinet in 1997
https://en.wikipedia.org/wiki/Blair_ministry
This leaves 1 Home Secretary only that was in both cabinets. Initials JS
It is now up to you to determine the rest and check that he was an ever-present.
Like Captain, I do not understand the latter part of your question.