  Symmetric Teleparallel Gravity allows for the reformulation of gravity in the
form of nonmetricity by vanishing the contorsion term in the generic affine
connection. Our focus is on investigating a recently proposed extension of this
theory in which the Lagrangian has the form $f(Q,C)$ by incorporating the
boundary term $C$. In this work, we first use a reconstruction approach in
$f(Q,C)$ gravity that might admit the $\Lambda$CDM expansion history.
Furthermore, we perform a novel approach for cosmological reconstruction of
$f(Q,C)$ gravity in terms of e-folding, and it shows how any FLRW cosmology can
arise from a specific $f(Q,C)$ gravity. A variety of instances are provided
using this approach in which $f(Q, C)$ gravity is reconstructed to yield the
well-known cosmic evolution: $\Lambda$CDM era, acceleration/deceleration era
which is equivalent to the presence of phantom and non-phantom matter,
late-time acceleration with the crossing of phantom-divide line and transient
phantom era.
