Context : Morphology is the most accessible tracer of galaxies physical structure , but its interpretation in the framework of galaxy evolution still remains problematic .
Its quantification at high redshift requires deep high-angular resolution imaging , which is why space data ( HST ) are usually employed .
At z > 1 , the HST visible cameras however probe the UV flux , which is dominated by the emission of young stars , which could bias the estimated morphologies towards late-type systems .

Aims : In this paper we quantify the effects of this morphological k-correction at 1 < z < 2 by comparing morphologies measured in the K and I-bands in the COSMOS area .
The Ks-band data indeed have the advantage of probing old stellar populations in the rest frame for z < 2 , enabling determination of galaxy morphological types unaffected by recent star formation .

Methods : In paper I we presented a new non-parametric method of quantifying morphologies of galaxies on seeing-limited images based on support vector machines .
Here we use this method to classify \sim 50 000 Ks selected galaxies in the COSMOS area observed with WIRCam at CFHT .
We use a 10-dimensional volume , including 5 morphological parameters , and other characteristics of galaxies such as luminosity and redshift .
The obtained classification is used to investigate the redshift distributions and number counts per morphological type up to z \sim 2 and to compare them to the results obtained with HST/ACS in the I-band on the same objects .
We associate to every galaxy with Ks < 21.5 and z < 2 a probability between 0 and 1 of being late-type or early-type .
We use this value to assess the accuracy of our classification as a function of physical parameters of the galaxy and to correct for classification errors .

Results : The classification is found to be reliable up to z \sim 2 .
The mean probability is p \sim 0.8 .
It decreases with redshift and with size , especially for the early-type population , but remains above p \sim 0.7 .
The classification globally agrees with the one obtained using HST/ACS for z < 1 .
Above z \sim 1 , the I-band classification tends to find less early-type galaxies than the Ks-band one by a factor \sim 1.5 , which might be a consequence of morphological k-correction effects .

Conclusions : We argue therefore that studies based on I-band HST/ACS classifications at z > 1 could be underestimating the elliptical population .
Using our method in a K _ { s } \leq 21.5 magnitude-limited sample , we observe that the fraction of the early-type population is ( 21.9 % \pm 8 % ) at z \sim 1.5 - 2 and ( 32.0 % \pm 5 % ) at the present time .
We will discuss the evolution of the fraction of galaxies in types from volume-limited samples in a forthcoming paper .