The Board of Internal Economy does. I haven't heard of public accounts being involved at all, but that doesn't mean they aren't. It may just be that I'm ill-informed. Definitely, the Board of Internal Economy is involved.
The trouble, in a nutshell, is that the Board of Internal Economy can't report back to the House. Ultimately, it's the House itself that would want to assert—I hesitate to use the word “control”, because we're talking about a building that's shared with the Senate—the kind of oversight that lets us say definitively that we want this feature to be present, or we are profoundly unhappy with the timeline that's been proposed, or that cost structure needs to get signed off by somebody. It has to be the House of Commons as a whole.
The trouble is that the board can't report to the House. We can report to the House. It could be any committee, but it needs to be a committee of Parliament—an actual committee, as opposed to the board—doing the detail work of hearing the witnesses, keeping track of the changes from one year to the next through multiple Parliaments, because the Centre Block won't be finished until multiple Parliaments have gone through, and potentially multiple changes of ministers and even possibly governments.
All we can do is report to the House. Then the report can be debated and potentially enacted, and it becomes a House order to the people who are actually doing this work on our behalf. That's the logic of it.
Does that answer your question?