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Let $p$ be a normalized (monic and centered) quartic polynomial withnon-trivial symmetry groups. It is already known that if $p$ is unicritical,with only two distinct roots with the same multiplicity or having a root at theorigin then the Julia set of its Chebyshev's method $C_p$ is connected andsymmetry groups of $p$ and $C_p$ coincide~[Nayak, T., and Pal, S., Symmetry anddynamics of Chebyshev's method, \cite{Sym-and-dyn}]. Every other quarticpolynomial 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 the Chebyshev'smethod $C_a$ of $p_a$ are investigated in this article for all real $a$. It isproved that all the extraneous fixed points of $C _a$ are repelling which givesthat there is no invariant Siegel disk for $C_a$. It is also shown that thereis no Herman ring in the Fatou set of $C_a$. For positive $a$, it is provedthat at least two immediate basins of $C_a$ corresponding to the roots of $p_a$are unbounded and simply connected. For negative $a$, it is however proved thatall the four immediate basins of $C_a$ corresponding to the roots of $p_a$ areunbounded and those corresponding to $\pm i\sqrt{|a|}$ are simply connected.

On dynamics of the Chebyshev's method for quartic polynomials