I have a feeling about your problem, but there may be a couple of things happening. Pulling that battery down to 4.5-volts says there is a LOT of current being drawn from it.
I am going to assume the 4.5 volts is across the battery posts, not the clamps. Just a little corrosion under the clamps can add a lot of drop during cranking. Just for kicks, measure the drop from the center of the post to the battery clamp. That should be no more than a tenth of a volt during cranking.
I am also assuming a good ground connection to the engine block or starter mounting bolts.
Voltage drop across the positive cable to the solenoid should only be a couple of tenths of a volt too. I am interested where the 2.5 volts (4.5 – 2.0) goes when you crank. The voltage drop across the solenoid should be only another couple of tenths. If you are seeing a couple of volts – you definitely have a bad solenoid.
Once you get a good solenoid in the Jeep, it’s back to looking at the starter. Pulling the battery down to 4.5 volts says there is a LOT of draw by that starter. You might have a short in the armature. If the starter does not stop on that set of shorted commutator bars, it will crank past the shorted bars. But, if the starter stops on a shorted set of commutator bars, the starter draws a bunch of current and won’t turn. If the starter does have an issue, it may have burned the contacts in the solenoid and it may be the root cause of the defective solenoid.
Taking the starter to a good auto electrical shop and having it checked would answer the question for you. Auto Zone and the like can’t test a starter like this one – it takes a specialized shop with an old guy to do it right.
You asked about a ground strap. Ground straps generally are used to make a good connection between the frame and the engine block where the battery negative terminal is connected. They are often just jumpered across an engine vibration isolator. Ground straps make lights and such work better, but they are not a part of this cranking problem.
Finding that lost 2.5 volts from the positive battery terminal to the starter post is going to go a long way toward fixing the problem.