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The drone doesn't have a full action unless the mechanic gives up some of their actions.
The rules suggest that it takes a full action to perform a crew action.
There is also the rules about class features.
Class features and items affect crew actions only if specifically noted in the class feature or item.
The drone is a class feature of the mechanic class. I can find nothing in their description indicating that the drone can perform a crew action.

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There is a discussion on this very topic over in rules area of Starfinder. Likely a better place as it is not a Society question.
Keep in mind that the drone is being controlled. It is not acting on it's own. So the mechanic would have to focusing on controlling the drone instead of the task at hand (engineering or science officer).
As pointed out by BretI, the rules do not allow for a drone to be part of starship combat.