  We calculate, at the one loop level, penguin diagrams for improved staggered
fermion operators constructed using various fat links. The main result is that
diagonal mixing coefficients with penguin operators are identical between the
unimproved operators and the improved operators using such fat links as Fat7,
Fat7+Lepage, $\bar{\rm Fat7}$, HYP (I) and HYP (II). In addition, it turns out
that the off-diagonal mixing vanishes for those constructed using fat links of
Fat7, $\bar{\rm Fat7}$ and HYP (II). This is a consequence of the the fact that
the improvement by various fat links changes only the mixing with higher
dimension operators and off-diagonal operators. The results of this paper,
combined with those for current-current diagrams, provide the complete matching
at the one loop level with all corrections of ${\cal O}(g^2)$ included.
