I understand what the job of a Head Coach is, but what does the Assistant Head Coach actually do during games/in the office?
Typically, in NFL and other sports, assistant coaches are responsible for a particular aspect of the team, while the head coach is responsible for the overall direction of the team.
For example in football you might have a quarterback coach, an offensive line coach, a running back coach, a kicking coach, a special teams coach, an athletic fitness coach, many of whom are given the title of assistant coach. Their job is to provide specific and expert direction in their areas. This isn't to say that the head coach wouldn't also be an expert in all areas of his team, because it would be bad if he wasn't. But his level of expertise will not be that of the specific coaches, unless he happened to grow out of that position or coaching position before being promoted to head coach.
It is very common for someone to be promoted to head coach and keep the responsibilities he held before promotion (especially in interim promotions).
Some teams do have a specific title of "Assistant Head Coach" - their responsibility is to act as the enforcers of the plan set forth by the head coach. The head coach can only supervise so many things at a time - some clubs feel the need for the extra management. Only two NFL clubs have an "Assistant Head Coach" position that is not also another type of coach like Offensive Line or Defensive Line - Kansas City and St. Louis. It could be a regional thing, both those teams have had a pretty rocky couple years lately, so it could just be that they wanted the extra emphasis on following the team direction.
I can't find a reference for it, but I would assume that for the teams that have Assistant Head Coaches that are also another type of coach (about half of them do) that if for some reason the Head Coach were unavailable, incapacited, or otherwise unable to perform his duties, that the Assistant Head Coach would take his position.
The duties of an assistant head coach vary widely, because each assistant coach on a football team takes on responsibilities that are designated by the head coach of the team. As an analogy of how duties might differ depending on the head coach, many coordinators may have a title, but plays might actually be called by the head coach (due to the head coach's previous offensive or defensive acumen), while other coordinators have handle play-calling, subject to the strategy set by the head coach.
As @corsiKa has mentioned, many assistant head coaches can be position coaches, while some coordinators may also serve in this role.
From general observation, I've seen that head coaches or team owners might designate an assistant head coach for several reasons:
- Designate a successor (implied in the case of the Dallas Cowboys and Jason Garret, though it is against league rules)
- Reward a long-time/loyal/key assistant with a more prestigious title (particularly if not already a coordinator) - seen at many colleges and the pros (see Rod Marinelli at several college/NFL teams, Brad Seely as special teams coach/asst head coach at several NFL teams, and Mike Singletary and the Minnesota Vikings as another example)
- Provide justification for higher pay for a non-coordinator (see the Jason Garret case)
While coordinators sometime serve as assistant head coaches, quite frequently special teams or position coaches are rewarded with this title/role. Not all teams have the asisstant head coach role.