  We present ASCA observations of LINERs without broad H$\alpha$ emission in
their optical spectra. The sample of "type 2" LINERs consists of NGC 404, 4111,
4192, 4457, and 4569. We have detected X-ray emission from all the objects
except for NGC 404; among the detected objects are two so-called transition
objects (NGC 4192 and NGC 4569), which have been postulated to be composite
nuclei having both an HII region and a LINER component. The images of NGC 4111
and NGC 4569 in the soft (0.5-2 keV) and hard (2-7 keV) X-ray bands are
extended on scales of several kpc. The X-ray spectra of NGC 4111, NGC 4457 and
NGC 4569 are well fitted by a two-component model that consists of soft thermal
emission with $kT\sim0.65$ keV and a hard component represented by a power law
(photon index $\sim$ 2) or by thermal bremsstrahlung emission ($kT\sim$ several
keV). The extended hard X-rays probably come from discrete sources, while the
soft emission most likely originates from hot gas produced by active star
formation in the host galaxy. We have found no clear evidence for the presence
of active galactic nuclei (AGNs) in the sample. If an AGN component is the
primary ionization source of the optical emission lines, then it must be
heavily obscured with a column density significantly larger than $10^{23}$
cm$^{-2}$. Alternatively, the optical emission could be ionized by a population
of exceptionally hot stars.
