No. The grouped unit that takes the defeated unit's place continues its turn from where the previous unit left off. A group can collectively only attack once per turn, which means it does not get to attack if the previous unit did.
This applies even if the previous unit had a talent that altered its attack timing. For example, if an harrier uses Quick Strike before spires fire, and is then defeated by a spire, the next unit in the group cannot attack during the normal attack phase because the group already attacked that turn.
However, the new unit will still retaliate from the first attack against it, even if the unit that was previously on top already retaliated.