  There is no supercyclic power bounded operator of class
$C_{1{\textstyle\cdot}}.$ There exist, however, weakly l-sequentially
supercyclic unitary operators$.$ We show that if $T$ is a weakly l-sequentially
supercyclic power bounded operator of class $C_{1{\textstyle\cdot}}$, then it
has an extension $\widehat T$ which is a weakly l-sequentially supercyclic
singular-continuous unitary (and $\widehat T$ has a Rajchman scalar spectral
measure whenever $T$ is weakly stable)$.$ The above result implies
$\sigma_{\kern-1ptP}(T)=\sigma_{\kern-1ptP}(T^*)=\varnothing$, and also that if
a weakly l-sequentially supercyclic operator is similar to an isometry, then it
is similar to a unitary operator.
