Well for starters this assumes the extended object has a rest frame to begin with. If the constituents of the object are moving radially relative to one another, such as in an arbitrary fluid, then it doesn't make much sense to talk about the rest frame of the object itself, right? In fact in general the best we can do is talk about the individual rest frames of the constituents of the objects. In a fluid these individual rest frames would correspond to those of the individual fluid elements. When can we talk about the global rest frame of an extended object itself? Precisely when the object is rigid.
But the definition of rigidity in relativity is quite different from the definition of rigidity in Newtonian mechanics. The precise definition requires some mathematical machinery but intuitively rigidity in relativity, often termed Born rigidity, means that given any constituent of the extended object, the spatial distances in the rest frame of this constituent to all neighboring constituents of the object remain constant i.e. there are no relative radial velocities between the neighboring constituents of the object. Note that this definition still allows a rigidly rotating object to have an extended rest frame.
Now coming back to your question, imagine we have a rod at rest in an inertial frame and say we want to accelerate the rod along its length. For the reasons explained above, if we even want to talk about the rest frame of the entire rod itself during the acceleration phase, we better make sure the acceleration is done in a Born rigid manner i.e. the rod must remain rigid during the acceleration phase. Then we can indeed talk about the global rest frame of the rod i.e. an extended frame in which all points of the rod are at rest. In order to do this, each point of the rod must receive a different proper acceleration-in fact we must impart a proper acceleration to each point of the rod inversely proportional to the fixed spatial location of the point in the rest frame of the rod. Now obviously in this case if all points of the rod are accelerated simultaneously in the inertial frame in which the rod is initially at rest then all points of the rod will accelerate simultaneously in the rest frame of the rod by construction*
On the other hand, say we don't accelerate the rod Born rigidly. Say we impart to all points of the rod the same proper acceleration and do so simultaneously in the inertial frame in which the rod is initially at rest. Then in fact it no longer makes sense to talk about the extended rest frame of the rod itself because in the individual rest frames of the points of the rod, the neighboring points will have radial velocities. Furthermore, now if we go to the rest frame of any given point of the rod, then relative to this frame the other points of the rod will have actually accelerated at different times i.e. even though all points were accelerated simultaneously in the initial inertial frame, because we gave all points the same proper acceleration we find that in the rest frame of any given point of the rod the other points will not have accelerated simultaneously. This is in fact the content of the famous Bell spaceship paradox:
http://en.wikipedia.org/wiki/Bell_spaceship_paradox
*We have implicitly assumed here that we can actually talk about global Einstein simultaneity in the extended rest frame of the rod. Rigidity is actually not enough to guarantee this. The reason we can make this assumption is due to the irrotationality of the rod motion.
