In the Overmind Player Phase, or Overmind/Alien Activation Phase, the Overmind Player gets to Activate each of its Alien Units once, except such Alien Units that have already had an Activation during the Hero Player Phase.
At the beginning of the Overmind Player Phase, before Activating anything, the Overmind Player may first choose to discard up to two Mutation cards from their hand, and then draws Mutation cards until they have five of them in hand. (The rulebook gives somewhat conflicting instructions here; page 13 says to draw 5 cards in the first Round, and 2 cards in later rounds, while page 14 says optionally discard then draw until 5 cards in hand.)
Activation
An Activation normally consists of selecting a Hero or Alien Unit and taking up to two Actions with it. Each Hero and each Alien Unit can only be activated once during a Round, unless some special ability says otherwise.
When a Hero, Alien Unit or Blip Token is Activated, discard any unused Overwatch token it may have, then take two Actions. The actions available are Move, Sprint, Melee Attack, Shoot Attack, Interact, Overwatch, and for Blip Tokens: Flip Blip Token.
Note: an out-of-turn Activation still counts towards the once-per-turn limit, but a triggered Action that happens without an Activation does not. Consequently, for example, the Overmind Player's Interruption ability, which lets it activate one Alien Unit during the Hero Players's Turn, means the Alien Unit Activated by that ability can't be Activated next time it becomes the Overmind Player's Turn; but Overwatch attacks and Alien Behaviour Actions don't count as Activations.