Integrity builds trust. A lack of trust will lead to a diminishing of influence. This one has given me some problems. Fame is composed of many elements, rank reputation, name recognition, achievement and accomplishment as well, and there are others. Fame can be a component in influence.
Fame is certainly an aid to charisma.
Certainly for fame to be of value it must be positive fame. Notoriety of the negative variety will cause many problems and must be avoided. Achievement can be a window into a person's heart, showing us what they hold important and their dedication to those goals. It does not matter whether the accomplishments are in combat, the arts or service. A desire to achieve is not enough.