What do you get for winning a Challenge Head of Faith war in CK3?

The Challenger Head of Faith becomes Head of Faith (patch 1.20.0.3).

What does the Challenge Head of Faith casus belli cost?

piety; the amount depends on the titles, claims and distance, worked out when you declare.

Casus belli · Holy wars

Challenge Head of Faith

The war is called “Challenge Head of Faith”

Challenge Head of Faith is a casus belli in Crusader Kings III (holy wars). If the attacker wins: The Challenger Head of Faith becomes Head of Faith. If the defender wins: Your challenge against the Head of Faith fails. Declaring it costs piety. The war goal is worth 80% war score. Patch 1.20.0.3, from game/common/casus_belli_types/pam_wars.txt.

Quick answers (3)

What do you get for winning a Challenge Head of Faith war in CK3?: The Challenger Head of Faith becomes Head of Faith (patch 1.20.0.3).
What happens on a white peace?: The defender stays the Head of Faith.
What does the Challenge Head of Faith casus belli cost?: piety; the amount depends on the titles, claims and distance, worked out when you declare.

How the war ends

Attacker wins

The Challenger Head of Faith becomes Head of Faith.

White peace

The defender stays the Head of Faith.

Defender wins

Your challenge against the Head of Faith fails.

If the war becomes invalid

The war ended: the defender is no longer Head of Faith.

Who can declare it

The game shows no text of its own for these conditions; the script is below.

Every holy wars war (the group's allowed_for_character) (24 lines of game script)

OR = {
	top_liege = this
	any_liege_or_above = {
		faith = prev.faith
		count = all
	}
}
faith = {
	NOR = {
		has_doctrine_parameter = unreformed
		AND = {
			has_doctrine_parameter = holy_wars_forbidden
			NOT = {
				prev = {
					has_character_flag = bonus_major_holy_war
				}
			}
		}
	}
}
herders_and_tributary_constraints = yes
pam_personal_pacifism_offensive_war_ban_trigger = yes
tgp_dynastic_cycle_offensive_wars_ban_trigger = yes
tgp_war_ban_all_ministers_trigger = yes

As written in game/common/casus_belli_groups/00_casus_belli_groups.txt.

Attacker (allowed_for_character) (4 lines of game script)

scope:attacker = {
	is_landed = yes
	pam_can_use_challenge_hof_cb_trigger = yes
}

As written in game/common/casus_belli_types/pam_wars.txt.

Defender (allowed_against_character) (10 lines of game script)

scope:defender = {
	is_landed = yes
	faith = scope:attacker.faith
	scope:attacker.faith.religious_head ?= this
}
scope:attacker = {
	NOT = {
		target_is_liege_or_above = scope:defender
	}
}

As written in game/common/casus_belli_types/pam_wars.txt.

Cost to declare

Costs piety. The amount is worked out when you declare (by the titles targeted, claims, distance and more), so the script is shown as written:

Cost (1 line of game script)

piety = major_piety_value

As written in game/common/casus_belli_types/pam_wars.txt.

Source: game/common/casus_belli_types/pam_wars.txt and the English localisation, Crusader Kings III 1.20.0.3. “The attacker”, “the defender” stand where the game names the character; “…” marks other words it fills in during play.