[PvK]

Are you asking about Star Trek, TNG, SE4, Earth navies, or what?

Task force suggests a force assembled for a specific task, usually with multiple types of forces, and can be mixed branches (e.g. naval plus marines) as well.

Battle group similarly would vary - it just implies a group worthy of fighting a "battle" by itself.

Assault fleet, spelled correctly, implies a force specifically for an assault, which might tend to mean landing an invasion force.

Fleet could mean either a group as in SE4, or all of the ships of a nation, or all of the ships of a nation in a particular theatre (e.g. Baltic Sea Fleet) or at a location (e.g. Earth Defense Fleet).

Military nomenclature likes to pretend to be authoritative and meaningful, whilst at the same time meaning very different things in different times and places. A lot of military culture and language has to do with insisting on something absurd and inconsistent, and insisting it go unquestioned.

Captain is generally a naval rank of the commander of a ship of any size.

Fleet captain is something specific, and could mean different things.

Commodore is a higher rank than captain, and is generally a rank of someone who commands a group of vessels, or is responsible for a region (i.e. in charge of the vessels which are assigned to a region). A commodore's flag (HQ) might be located at a base or on a ship, and can move. Sometimes if a flagship is damaged, a commodore (or admiral or whover is in charge) will "transfer his flag" to another ship, generally supplanting the captain of that ship at least for major decisions.

Rear Admiral in the US navy can either be sort of like a commodore (just a rank higher than captain and higher than commodore) or can also be someone higher than such people. Those are called "lower half" and "upper half" respectively, but I doubt that halfsies silliness gets absorbed into Star Trek.

Admiral used to be sufficient to describe the commander of a whole navy. It can also be a rank above Vice Admiral, and in general means someone of very high rank in the navy, who may move between HQ assignments, and is likely to be temporarily assigned as the commander of a large and important operation, or be in command of a regional fleet, or something.

Fleet Admiral in the US navy means the highest ranking admiral in the entire nation's navy. Commander in chief of the navy.

PvK