Can someone explain the actual mechanism behind the F-wave? Is it really different from a H-reflex response or does it just have another name for historical reasons? The H-reflex (Hoffmann's reflex) is a reflex motor response triggered by a afferent sensory nerve signals. As far as I could find, the literature describes the F-Wave as a motor response elicited by "backfiring" motor fibres that were antidromically activated by stimulation. It is unclear to me how such backfiring is coherent with the concepts of refractory periods and normal axonal signal transmission.
This is a great question, and I've had trouble finding any authoritative study; since F-waves are mostly used as a diagnostic tool it doesn't seem like they've been studied much at the mechanistic level. The only non-clinical usages seem to be in pharmacology experiments in intact organisms (in particular people) where people aren't going to be going in and doing whole-cell patch clamp in the spinal cord.
Balbi, P., Martinoia, S., & Massobrio, P. (2015). Axon-somatic back-propagation in detailed models of spinal alpha motoneurons. Frontiers in computational neuroscience, 9, 15.
has some relevant simulations, though they don't entire answer your question. However, I think their results are sufficient to make some good guesses. Without developing my own simulations, though, I'll have to mostly hand-wave after establishing some key facts.
Axons are small, so it doesn't take that many ions to depolarize a stretch of axon to threshold. The axon hillock has a lot of sodium channels to sort of 'jump-start' the action potential down the axon, but that's in a context where the rest of the soma is already depolarized.
In a back-propagating action potential, the ions flowing into small compartments eventually meet the soma: the soma is big, and so the number of ions needed to depolarize the soma to threshold is large (lots of capacitance in the membrane to charge): the axon hillock may not immediately bring the soma to threshold. This is called "impedance mismatch" and is also important at synapses (where axon terminals are bigger than the axons).
And so now for the hand-waving component... I can't say for certain which channels are involved, but it's possible that the antidromic action potential depolarizes the soma just a little bit, enough to open some voltage-gated channels but few enough that it takes some time for the rising phase to occur. Could involve other ions that aren't sodium, like calcium; could involve the proximal dendrites as well. Because of the different voltage sensitivities of different channels in different compartments, by the time the whole soma depolarizes sufficiently, the proximal part of the axon has had time to recover enough that it can spike again.
One other piece of evidence in support of this mechanism is that the number of cells that fire an orthodromic spike in the F-wave after the antidromic spike is few and variable - that is, it's a small fraction and that fraction varies from pulse to pulse. That seems to suggest that the specific conditions of the soma and nearby processes matters, so there must be a fairly narrow range of parameters that allows an orthodromic spike to follow.
H-reflex is a reflectory reaction of muscles triggered by stimulation of sensory (afferent) nerve fibres type Ia. Stimulation should have short duration and small amplitude.
When stimulation is supra-maximal, we can observe antidromic action potential through the motor (efferent) nerve fibres ($\alpha$). This is F-wave.