Ramified frobenius has no canonical lift

Given: The data and hypotheses of the statement.

Gaussian and eisenstein frobenius: In $\mathbb{Q}(i)$, an odd prime p splits if $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and is inert if $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$; arithmetic Frobenius sends $i\mapsto i^p$. The prime 2 ramifies. In $\mathbb{Q}({\zeta }_3)$, a prime $p\mathrm{\ne }3$ splits if $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$ and is inert if $p\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$; arithmetic Frobenius sends ${\zeta }_3\mapsto {\zeta }_3^p$. The prime 3 ramifies.

Arithmetic frobenius coset: For finite Galois L/K and nonzero $P\mid p$, the arithmetic Frobenius coset is the unique element of $D(P\mathrm{/}p)\mathrm{/}I(P\mathrm{/}p)$ corresponding under the residue isomorphism to $x\mapsto x^{Np}$ on $\kappa (P)$, where $Np=\mathrm{\mid }\kappa (p)\mathrm{\mid }$. It is defined also when P is ramified. Its inverse is called geometric Frobenius. The quotient element is distinguished; a representative in D need not be unique.