  Both 4-harmonic and ES-4-harmonic maps are two higher order generalizations
of the well-studied harmonic map equation given by a nonlinear elliptic partial
differential equation of order eight. Due to the large number of derivatives it
is very difficult to find any difference in the qualitative behavior of these
two variational problems. It is well known that the small hypersphere
$\iota\colon\mathbb{S}^m(\frac{1}{\sqrt{4}})\to\mathbb{S}^{m+1}$ is a critical
point of both the 4-energy as well as the ES-4-energy but up to now it has not
been investigated if there is a difference concerning its stability.
  The main contribution of this article is to show that the small hypersphere
is unstable with respect to normal variations both as 4-harmonic hypersphere as
well as ES-4-harmonic hypersphere and that its normal index equals one in both
cases.
