  Motivated by the recent works on the stability of symmetric periodic orbits
of the elliptic Sitnikov problem, for time-periodic Newtonian equations with
symmetries, we will study symmetric periodic solutions which are emanated from
nonconstant periodic solutions of autonomous equations. By using the theory of
Hill's equations, we will first deduce in this paper a criterion for the
linearized stability and instability of periodic solutions which are odd in
time. Such a criterion is complementary to that for periodic solutions which
are even in time, obtained recently by the present authors. Applying these
criteria to the elliptic Sitnikov problem, we will prove in an analytical way
that the odd $(2p,p)$-periodic solutions of the elliptic Sitnikov problem are
hyperbolic and therefore are Lyapunov unstable when the eccentricity is small,
while the corresponding even $(2p,p)$-periodic solutions are elliptic and
linearized stable. These are the first analytical results on the stability of
nonconstant periodic orbits of the elliptic Sitnikov problem.
