Describe the proposed counterattack in more detail. Let's assume the garrison has surrendered and the FJ holds the north shore, with heavy air activity and some over the beach logistics occurring.
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After the shock of the Fall of France, the invasion expected by many occurred, with a night paratroop landing on the Isle of Wight on the 15 July 1940.
This landing, timed for the full moon of 19 July, triggered a full on invasion scare with British defense plans put into motion to repel a full scale invasion.
The expected follow-up invasion of England proper did not occur, but the British preparations to defend the rest of the UK did give the German troops on the Isle of Wight 72 hours or so to consolidate their control over the island and conduct limited over-the-beach resupply. This resupply involved limited reinforcement from the 22 Airlanding Division, however great difficulties were found unloading artillery from small boats over beaches.
The focus of the Battle of Britain then became the Isle of Wight and the waters around it, with the British attempted to prevent the reinforcement of the German paratroops and destroying any German attempts to establish an airfield, and the Germans attempting to attrit the RN forces blockading their air-head while improving their rudimentary airfield to allow control of the island.
While at this time the RAF was inferior in the number of single-engine fighters, it was still at rough parity with the Lutwaffe's forces.
Once the British got over their initial shock, planning began for the counter-attack.
The geography of the Isle of Wight was the key to the plan, with the British planners correctly assessing that the German defenders would be without tanks and short on artillery.
While the Camden-Lloyd firm had experimented with amphibious tanks in the 1930s (1), none were immediately available. The firm was, however, contacted and conversion began, expected to be completed by September.
The Royal Marines were the obvious choice to lead the retaking of the island, and 101 Brigade began assembling. Lacking their own artillery, two Royal Artillery regiments were assigned. It was a critical feature of the upcoming battle that the 13 500 yard (12.2km) range of this artillery could reach much of the island from batteries established on the mainland.
Additionally, liaison officers were assigned from each of the RN ships in Portsmouth, with the plan being to use the RN as floating artillery batteries.
The preparatory stages of the battle were a task for the RAF and the Navy, with the Navy preventing major reinforcement of the island while the respective air forces fought a grinding battle of attrition over the Solent, the Channel and the Isle of Wight itself.
The Royal Artillery and Royal Marine Division was in place by the end of the month, and Army Cooperation squadrons hunted down the limited amount of German artillery for destruction by the RN or RAF. The FJs still had their mortars, but ammunition for them was limited.
When it happened, the invasion of the north-west corner of the island began on 2 August 1940, and the paralyzing affect of not only the British 25 pounders but also British destroyer fire prevented coordinated German defense. Yarmouth was the focus of the British landing, and the line Hamstead-Brightstone was established by the night of 2 August.
The remnants of the German forces defending Yarmouth retreated to the rubble that was the old fort at the Needles, and were destroyed by 15 inch gunfire by the old battleship Revenge.
Having absolute naval supremacy in the Solent, and having their artillery based on the mainland the British were able to use the small port of Yarmouth solely to resupply the Marines and evacuate casualties. The Germans had no such luxury, with the RAF and RN competing to attack any resupply vessels, and then the supplies moved across the island. The Luftwaffe could, of course, extract a price from this but they could not prevent losses of shipping.
German counterattacks on the night of 2-3 August were broken up by British artillery fire - it was later found that 25% of all British 25 lber ammo had been stockpiled in Portsmouth for support of this operation, allowing rapid fire of 4 rpm to be maintained for longer than the Germans could sustain a counterattack.
British troops pushed along the north coast to Cowes, forcing the German defenders to decide whether to stay in place and be outflanked or maneuver and be caught outside their foxholes by the British artillery.
Once Cowes fell to the Marines on 5 August, the fate of the defenders was sealed.
The last holdouts surrendered on 12 August.
The first British amphibious tanks landed on 28 August, having successfully navigated the crossing from Lymington to Yarmouth.
https://en.wikipedia.org/wiki/Vickers-Carden-Loyd_Light_Amphibious_Tank
http://www.visitmyharbour.com/harbours/solent/yarmouth/charts.asp?cattype=5