Let p be a normalized (monic and centered) quartic polynomial with non-trivial symmetry group. It is already known that if p is unicritical, with only two distinct zeros with the same multiplicity or having a root at the origin then the Julia set of its Chebyshev’s method \(C_p\) is connected and symmetry groups of p and \(C_p\) coincide (Nayak, T., Pal, S, Mediterr. J.Math. 22(1), 12 (2025)). Every other quartic polynomial is shown to be of the form \(p_a (z)=(z^2 -1)(z^2-a)\) where \(a \in {\mathbb {C}}\setminus \{-1,0,1\}\) . Some dynamical aspects of Chebyshev’s method \(C_a\) of \(p_a\) are investigated in this article for all real a. It is proved that all the extraneous fixed points of \(C _a\) are repelling, which gives that there is no invariant Siegel disk for \(C_a\) . It is also shown that there is no Herman ring in the Fatou set of \(C_a\) . For positive a, it is proved that at least two immediate basins of \(C_a\) corresponding to the zeros of \(p_a\) are unbounded and simply connected. For negative a, it is however proved that all the four immediate basins of \(C_a\) corresponding to the zeros of \(p_a\) are unbounded and those corresponding to \(\pm i\sqrt{|a|}\) are simply connected.
