theoretical situation

[TStoneMBD]

You open on the button with K8o. SB and BB both call.

Flop is Ad5d5h

Both players check to you and you check. (lets not talk about this flop check its irrelevant, lets just assume its good.)

Turn is a Ks

1) SB bets, BB folds

What is your action? Against what players do you raise for a free showdown and against what players do you call down? Do you ever fold?

Are there players who will fold a king to a turn raise in this spot? That's why I ask if a free showdown raise has merit against certain players in these situations.

2) SB checks, BB bets

Same question