If the target units remain in cover at the end of their counter-charge, AFAIK they still get the cover save.
Put another way, why would they lose their cover save under these circumstances?

. Infantry units from formations taking a charge
action may not take cover saves (they are assumed to have left
cover to charge the enemy), but other infantry units
may take cover saves normally.

This does pop up often and I believe has been added to the FAQ updates for better wording.
I think the main problem comes from players mis-reading the rules.
I've had games where players claim the rules say no cover save for "a charge move" and believe a counter charge comes under this move but the rules are actually no cover saves for "Taking a charge action" with the charge action being the engage action and defenders do not take an engage (or charge) action.

From 1.12.5

I think that if you say "charge action" it's much more likely mean "engage action" than "charge move".

The distinction between actions and moves is pretty clear and never used interchangeably, whereas people often say "my tacticals are going to charge your scouts" when they declare an action.

Indeed. Earlier bits of the assault rules make quite a big thing of distinguishing between the two phrases:

'1.12.3 Make Charge Move

A formation undertaking an engage action is allowed to make one move (not a double distance move as is the case in many sets of wargame rules, not least many Games Workshop games), and then fights an assault against the enemy formation that was chosen as the target of the charge. This move is known as the charge move.'

Emphasis not added by me.

So 1.12.5 contains a typo one way or another, either it's 'charge move' or 'engage action'. Given that the justification is leaving cover to charge the enemy I still think 'charge move' makes more sense in the context.

Also, if it's 'engage action' you get the absurd situation where an engaging formation that didn't move at all (already in firefight range) doesn't get cover saves while units counter-charging 10cm get cover saves. That's the kind of rule that destroys suspension of disbelief during a game.

Q: Is a formation that is allowed to move as part of an action, but which
decides to remain stationary, considered to be moving for the purposes of
triggering Overwatch fire? For example, if I take an Engage action, but don’t
move any units as they are already within firefight range of the enemy, can
my opponent make an Overwatch attack before the Assault takes place?
A: Yes, to both the question and the example. Overwatch is triggered when an
enemy formation “completes a move”. Formations that could move but
remain stationary have still “completed a move” and are therefore eligible
targets.

Q: One of the FAQ entries draws a conclusion that seems overly broad to me.
Section 1.7.1 talks about various orders allowing formations to move. The
Overwatch section of the FAQ rules that any time a formation is allowed to
move, it must move.
A: You can move zero, if you wish, but that is still considered a move. You
don’t have to move. In fact, some orders like Sustained Fire require you to
stay stationary. Other than that order, you are merely moving zero.
Withdrawls are different, in that they are optional movement. If you choose
not to move, you have not moved and do not allow a formation in
overwatch to activate against you.
When a unit does break, it does choose where and if to withdraw. I
personally advance my broken formations. There is a chance they may
unbreak next turn and it provides consternation to my opponent. However,
if you end your move (and yes, you can still move zero) within 15 cms of an
enemy, the unit is destroyed.