  A comprehensive extension of the ordinary 2-Higgs Doublet Model (2HDM),
supplemented by Vector-Like Quarks (VLQs), in the ``alignment limit'' is
presented. In such a scenario, we study the possibility that Large Hadron
Collider (LHC) searches for VLQs can profile their nature too, i.e., whether
they belong to a singlet, doublet, or triplet representation. To achieve this,
we exploit both Standard Model (SM) decays of VLQs with top-(anti)quark
Electromagnetic (EM) charge ($T$), i.e., into $b,t$ quarks and $W^\pm, Z,h$
bosons (which turn out to be suppressed and hence $T$ states can escape
existing limits) as well as their exotic decays, i.e., into $b,t$ (and possibly
$B$) quarks and $H^\pm, H, A$ bosons. We show that quite specific decay
patterns emerge in the different VLQ representations so that, depending upon
which $T$ signals are accessed at the LHC, one may be able to ascertain the
underlying Beyond Standard Model (BSM) structure, especially if mass knowledge
of the new fermionic and bosonic sectors can be inferred from (other) data.
