Should abortions of any kind be permitted?

No. At least from a Western perspective, unsafe, forced, criminal (eg., kicking or shooting the fetus), nor coerced abortions should never be allowed. Beyond that, it is a subjective matter. Late term abortions are distasteful and morally repugnant to many, but they are legal in a number of places, and they may prevent a lifetime of misery for the possible offspring.