  We develop a method to build new 5D $\mathcal{N}=1$ gauge models based on
Sasaki-Einstein manifolds $Y^{p,q}.$ These models extend the standard 5D ones
having a unitary SU$\left( p\right) _{q}$ gauge symmetry based on $% Y^{p,q}$.
Particular focus is put on the building of a gauge family with symplectic
SP$\left( 2r,\mathbb{R}\right) $ symmetry. These super QFTs are embedded in
M-theory compactified on folded toric Calabi-Yau threefolds $%
\hat{X}(Y^{2r,0})$ constructed from conical $Y^{2r,0}$. By using
outer-automorphism symmetries of 5D $\mathcal{N}=1$\textbf{\ }BPS quivers with
unitary SU$\left( 2r\right) $ gauge invariance, we also construct BPS quivers
with symplectic SP$\left( 2r,\mathbb{R}\right) $ gauge symmetry. Other related
aspects are discussed. Keywords: SCFT$_{5}$, 5D $\mathcal{N}=1$ super QFT on a
finite circle, Sasaki-Einstein manifolds, BPS quivers, outer-automorphisms.
