  The flat torus ${\mathbb T}={\mathbb S}^1\left (\frac{1}{2} \right ) \times
{\mathbb S}^1\left (\frac{1}{2} \right )$ admits a proper biharmonic isometric
immersion into the unit $4$-dimensional sphere ${\mathbb S}^4$ given by $\Phi=i
\circ \varphi$, where $\varphi:{\mathbb T} \to {\mathbb S}^3(\frac{1}{\sqrt
2})$ is the minimal Clifford torus and $i:{\mathbb S}^3(\frac{1}{\sqrt 2}) \to
{\mathbb S}^4$ is the biharmonic small hypersphere. The first goal of this
paper is to compute the biharmonic index and nullity of the proper biharmonic
immersion $\Phi$. After, we shall study in the detail the kernel of the
generalised Jacobi operator $I_2^\Phi$. We shall prove that it contains a
direction which admits a natural variation with vanishing first, second and
third derivatives, and such that the fourth derivative is negative. In the
second part of the paper we shall analyse the specific contribution of
$\varphi$ to the biharmonic index and nullity of $\Phi$. In this context, we
shall study a more general composition $\tilde{\Phi}=\tilde{\varphi} \circ i$,
where $\tilde{\varphi}: M^m \to {\mathbb S}^{n-1}(\frac{1}{\sqrt 2})$, $ m \geq
1$, $n \geq {3}$, is a minimal immersion and $i:{\mathbb
S}^{n-1}(\frac{1}{\sqrt 2}) \to {\mathbb S}^n$ is the biharmonic small
hypersphere. First, we shall determine a general sufficient condition which
ensures that the second variation of $\tilde{\Phi}$ is nonnegatively defined on
$\mathcal{C}\big (\tilde{\varphi}^{-1}T{\mathbb S}^{n-1}\big )$. Then we
complete this type of analysis on our Clifford torus and, as a complementary
result, we obtain the $p$-harmonic index and nullity of $\varphi$. In the final
section we compare our general results with those which can be deduced from the
study of the equivariant second variation.
