We apply concepts and methods due to \textit{D. Krupka} [Arch. Math., Brno 14, 207-214 (1978; Zbl 0428.58002)] to find all second order natural transformations of symmetric affine connections on manifolds into metrics on linear frame bundles. The complete family of naturally resulting metrics (to a fixed symmetric affine connection) is a real vector space generated by some generalization of two ``classical'' liftings.