Let $(Xt)t=0\infty$ be an irreducible reversible discrete time Markov chain on a finite state space $\Omega $.Denoteitstransitionmatrixby$P$.Toavoidperiodicityissues(andthusensuringconvergencetoequilibrium)oneoftenconsidersthecontinuous-timeversionofthechain$(X_t^{\mathrm{c}}){t \ge 0} $ whose kernel is given by $H_t:=e^{-t}\sum_k (tP)^k/k! $.Anotherpossibilityistoconsidertheassociatedaveragedchain$(X_t^{\mathrm{ave}}){t = 0}^{\infty}$, whose distribution at time $t$ is obtained by replacing $P$ by $At:=(Pt+Pt+1)/2$. A sequence of Markov chains is said to exhibit (total-variation) cutoff if the convergence to stationarity in total-variation distance is abrupt. Let $(Xt(n))t=0\infty$ be a sequence of irreducible reversible discrete time Markov chains. In this work we prove that the sequence of associated continuous-time chains exhibits total-variation cutoff around time $tn$ iff the sequence of the associated averaged chains exhibits total-variation cutoff around time $tn$. Moreover, we show that the width of the cutoff window for the sequence of associated averaged chains is at most that of the sequence of associated continuous-time chains. In fact, we establish more precise quantitative relations between the mixing-times of the continuous-time and the averaged versions of a reversible Markov chain, which provide an affirmative answer to a problem raised by Aldous and Fill.