^ That was my impression too, partially from Kirk's statement to Christopher a century later. In the 22nd century the organization that operates the starships would be under (or inside) the space probe agency, and not the other way around. Nor would the two exist completely separately.
We know that Captain Archer worked for Admiral Forrest, after that things get fuzzy. At some point an organization named Starfleet encloses them both and the NX-01, but the details of 22nd century Starfleet are quite vague.
Was Forrest the only Admiral? And when he was killed a new "only Admiral" was promoted to that position?