There is a skill called "Fury of the Undeath". It allows you to activate your Reanimate in addition to its normal activation. Does this mean that by sacrificing an action you enable your Reanimate to get four actions?

Yes, it can move and attack on each activation, for a total of 2 moves & 2 attacks.

Or it can move four times…

Nope.

"When activated, a familiar may perform a move action following the same rules as heroes. Familiars treat any special terrain (other than obstacles) as water during their movement. The familiar may perform additional types of actions during its activation, as noted on its Familiar card. These additional actions, unless otherwise noted, may be performed in addition to the move action and may interrupt the move action in a way similar to hero move actions. If no such other actions are noted, then all the familiar can do on its turn is move up to its Speed value."

"This familiar [Reanimate] is treated as a figure but cannot recover any amount of Wounds. It may perform 1 attack action (Blue, Red) during its activation. Each attack made by this familiar gains: Surge: +1 Wound"

Familiars get 1 move action. Reanimate gets an added attack action. That is all.

Ups! That's good to know.

Spot on and well illustrated Triu.

Course, you only need to look at the zoombie monsters to realise the intention was to not have the reanimate performing two move actions either. Thematically accurate at least.