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To explore the difficulties of classifying actions with the tracial Rokhlin property using K-theoretic data, we construct two 
$\mathbb{Z}_{2}$ actions 
$\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2}$ on a simple unital AF algebra 
$A$ such that 
$\unicode[STIX]{x1D6FC}_{1}$ has the tracial Rokhlin property and 
$\unicode[STIX]{x1D6FC}_{2}$ does not, while 
$(\unicode[STIX]{x1D6FC}_{1})_{\ast }=(\unicode[STIX]{x1D6FC}_{2})_{\ast }$, where 
$(\unicode[STIX]{x1D6FC}_{i})_{\ast }$ is the induced map by 
$\unicode[STIX]{x1D6FC}_{i}$ acting on 
$K_{0}(A)$ for 
$i=1,2$.
