Build an "internal base", such that when opened flat, or "unfolded" completely, the height of the base side of the pyramid would be greater (by perhaps 1.5 inches, for example) than the unfolded height ("thickness") of the other four sides. The material used for this could be anything machinable (wood, fiberboard, etc), or 3-D printable.
The purpose of this platform is to house, contain, and conceal your electronics as well as the majority of your mechanical linkages. You might picture this platform as being machined on the bottom side to receive a servo, mounted either by chemical adhesive or mechanical fastener(s), as well as whatever driving circuitry and/or power source you may wish to include within this internal base.
Then you might imagine four drilled holes, each one bearing an opening at one side of the platform (the side corresponding to the pyramid-side which linkage extended through that individual hole will eventually operate). With a four-armed servo-horn spline-mounted and fastened to your servo, you might extend a length of rigid wire or thin rod ("push rods") through each base-hole, such that rotational motion of the servo horn is translated along a linear trajectory that is that wire-segment's length, then (by use of a PASSIVE hinging mechanism of your choosing) back into rotary motion of the pyramid-side about the axis (pin or pins) of a hinge securing each side to the base of the structure. The center-ends of your rods are attached to your servo, the outermost ends attached to your pyramid sides by way of any anchor the allows freedom of movement (cut-off servo horn arms work well here).
That's a lot of words, but I think it's a good design. Using this method, the interior space of your pyramid could be utilized for any purpose (storage, display case, etc), while almost all of the functional mechanism remains out of sight.
I would gladly draw it for you for three dollars and ninety-nine cents. Just kidding, if you need a drawing to visualize what I typed, let me know.
Best of fortune on your project.