  A strong connection between cluster algebras and representation theory was
established by the cluster category. Cluster characters, like the original
Caldero-Chapoton (CC) map, are maps from certain triangulated categories to
cluster algebras and they have generated much interest. Holm and J{\o}rgensen
constructed a modified CC map from a sufficiently nice triangulated category to
a commutative ring, which is a generalised frieze under some conditions. In
their construction, a quotient $K_{0}^{sp}(\mathcal{T})/M$ of a Grothendieck
group of a cluster tilting subcategory $\mathcal{T}$ is used. In this article,
we show that this quotient is the Grothendieck group of a certain
extriangulated category, thereby exposing the significance of it and the
relevance of extriangulated structures. We use this to define another modified
CC map that recovers the one of Holm--J{\o}rgensen.
  We prove our results in a higher homological context. Suppose $\mathcal{S}$
is a $(d+2)$-angulated category with subcategories
$\mathcal{X}\subseteq\mathcal{T}\subseteq\mathcal{S}$, where $\mathcal{X}$ is
functorially finite and $\mathcal{T}$ is $2d$-cluster tilting, satisfying some
mild conditions. We show there is an isomorphism between the Grothendieck group
$K_{0}(\mathcal{S},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$ of the
category $\mathcal{S}$, equipped with the $d$-exangulated structure induced by
$\mathcal{X}$, and the quotient $K_{0}^{sp}(\mathcal{T})/N$, where $N$ is the
higher analogue of $M$ above. When $\mathcal{X}=\mathcal{T}$ the isomorphism is
induced by the higher index with respect to $\mathcal{T}$ introduced recently
by J{\o}rgensen. Thus, in the general case, we can understand the map taking an
object in $\mathcal{S}$ to its $K_{0}$-class in
$K_{0}(\mathcal{S},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$ as a
higher index with respect to the rigid subcategory $\mathcal{X}$.
