If you move an AA gun into a friendly power’s territory, you place a control marker under it to indicate that you still own it (page 25). If an enemy power captures the territory, they gain control of the AA gun (page 20). If a friendly power then captures the territory, they gain control of the AA gun (also page 20). However, if a friendly power liberates the territory, ownership of the AA gun passes to the original controller of the territory (also page 20).
In addition, if a friendly power captures the territory rather than liberating it because the original power’s capital is under enemy control, the friendly power gains control of the AA gun. If the friendly power’s capital is subsequently liberated, control of the territory goes back to that power, but control of the AA gun remains with the capturing power, so a control marker must be placed under it. (All of this is also on page 20.)
Complicated enough? Here are a couple of examples:
Example 1: The UK moves an AA gun to Soviet-controlled Karelia. The AA gun belongs to the UK, so a UK marker is placed under it. Then Germany captures Karelia. Germany gets the AA gun, and the UK marker is removed. The USA then liberates Karelia, and the territory and the AA gun become controlled by the USSR.
Example 2: The UK moves an AA gun to Soviet-controlled Karelia. The AA gun belongs to the UK, so a UK marker is placed under it. Then Germany captures Karelia. Germany gets the AA gun, and the UK marker is removed. Germany also captures Russia. The USA then captures Karelia (not liberates, because the USSR’s capital is enemy-controlled), and the territory and the AA gun become controlled by the USA. The USSR then recaptures Russia, so Karelia passes from USA control to Soviet control, but the AA gun in Karelia is retained by the USA, and a USA control marker is placed under it.